How to summon a Turtle...

freeztar · March 31, 2007

Turtle, continue out to 33! Trust me, it's worth it! :P

I have three sheets of squabble that look like your scan. I'm still trying to get the swirl to stabalize, so to speak, but now my roomate wants to jam on violin and guitar, so off I go...

Turtle · March 31, 2007

Turtle, continue out to 33! Trust me, it's worth it! :P

I have three sheets of squabble that look like your scan. I'm still trying to get the swirl to stabalize, so to speak, but now my roomate wants to jam on violin and guitar, so off I go...

Acknowledged & Roger Wilco

1=1
2=2
3=3
4=3+1
5=5
6=5+1
7=5+2
8=8
9=8+1
10=8+2
11=8+3
12=8+3+1
13=13
14=13+1
15=13+2
16=13+3
17=13+3+1
18=13+5
19=13+5+1
20=13+5+2
21=21
22=21+1
23=21+2
24=21+3
25=21+3+1
26=21+5
27=21+5+1
28=21+5+2
29=21+8
30=21+8+1
31=21+8+2
32=21+8+3
33=21+8+3+1
34= ninth Fibonacci #
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Turtle · March 31, 2007

Turtle, continue out to 33! Trust me, it's worth it! ;)

I have three sheets of squabble that look like your scan. I'm still trying to get the swirl to stabalize, so to speak, but now my roomate wants to jam on violin and guitar, so off I go...

I am green with envy that you have jammin' buddies. :( But that 's a thread of a different color so on to the numbers.

We have DigBog's question simmered to a conjecture - herein referred to as 'the digbog conjecture' - all positive integers may be expressed as the sum of Fibonacci numbers (henceforth and until further notice and on my authority called a Fibonacci Sum').

Now I'm glad I used a jeweler's drill and not the brace & bit on Freezy's point about how many Fibonacci numbers any given number requires in its 'Fibonacci Sum'. If we assume (presume?) the Digbog Conjecture is true, then Freezy's question is interesting & worthy of some further sleuthing.

We already know an integer may have more than 1 Fibonacci Sum, and that the fewest Fibonacci numbers required is 2, so we likely want to look to see if there is a 'most' value for how many Fibonacci numbers are used in a Fibonacci Sum.

Piece of cheese. :cup: :shrug:

CraigD · March 31, 2007

I feel tantalizingly close to a proof of the “the digbog conjecture”, following the lines of the question “is there a Fibonacci number F(n) such that a number x, F(n-1)<x<F(n) can’t be expressed as a sum of non-repeating Fibonacci numbers?” – though in the realm of math intuition “tantalizingly close” sometimes means “not even on the right track”. ;)

A few observations:

The conjecture is false for some generalized Fibonacci sequences, such as ”non-minimal” n-fib sequences (that is, any that don’t start {0,1,1,2} or similarly), such as {1,3,4,7,11 …}
The conjecture appears true for any “minimal” n-fib sequence, such as the 3-fib sequence {0,0,1,1,2,4,7,11 …} or the 9-fib sequence {1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144,16272,32512,64960,129792}
It appears to always be possible to find the terms that sum to a number by “working backward”, choosing the greatest fib number not greater than the remainder of the number minus the previously chosen numbers. No “intelligence” is required to find the terms summing to any number.

and a conjecture of my own

No number less than F(n)-1 will be the sum of more terms than the number of terms summing to F(n)-1

Just to demonstrate the unimpressive fact that I have a computer and a programming system that allows me to crank out numbers with ease (and for the viewing convenience of anyone who doesn’t), here’s a list of 1 through 987 as sums of 2-fib numbers, with a bit of extra data at the end:

1=1 (1)
2=2 (1)
3=3 (1)
4=3+1 (2)
5=5 (1)
6=5+1 (2)
7=5+2 (2)
8=8 (1)
9=8+1 (2)
10=8+2 (2)
11=8+3 (2)
12=8+3+1 (3)
13=13 (1)
14=13+1 (2)
15=13+2 (2)
16=13+3 (2)
17=13+3+1 (3)
18=13+5 (2)
19=13+5+1 (3)
20=13+5+2 (3)
21=21 (1)
22=21+1 (2)
23=21+2 (2)
24=21+3 (2)
25=21+3+1 (3)
26=21+5 (2)
27=21+5+1 (3)
28=21+5+2 (3)
29=21+8 (2)
30=21+8+1 (3)
31=21+8+2 (3)
32=21+8+3 (3)
33=21+8+3+1 (4)
34=34 (1)
35=34+1 (2)
36=34+2 (2)
37=34+3 (2)
38=34+3+1 (3)
39=34+5 (2)
40=34+5+1 (3)
41=34+5+2 (3)
42=34+8 (2)
43=34+8+1 (3)
44=34+8+2 (3)
45=34+8+3 (3)
46=34+8+3+1 (4)
47=34+13 (2)
48=34+13+1 (3)
49=34+13+2 (3)
50=34+13+3 (3)
51=34+13+3+1 (4)
52=34+13+5 (3)
53=34+13+5+1 (4)
54=34+13+5+2 (4)
55=55 (1)
56=55+1 (2)
57=55+2 (2)
58=55+3 (2)
59=55+3+1 (3)
60=55+5 (2)
61=55+5+1 (3)
62=55+5+2 (3)
63=55+8 (2)
64=55+8+1 (3)
65=55+8+2 (3)
66=55+8+3 (3)
67=55+8+3+1 (4)
68=55+13 (2)
69=55+13+1 (3)
70=55+13+2 (3)
71=55+13+3 (3)
72=55+13+3+1 (4)
73=55+13+5 (3)
74=55+13+5+1 (4)
75=55+13+5+2 (4)
76=55+21 (2)
77=55+21+1 (3)
78=55+21+2 (3)
79=55+21+3 (3)
80=55+21+3+1 (4)
81=55+21+5 (3)
82=55+21+5+1 (4)
83=55+21+5+2 (4)
84=55+21+8 (3)
85=55+21+8+1 (4)
86=55+21+8+2 (4)
87=55+21+8+3 (4)
88=55+21+8+3+1 (5)
89=89 (1)
90=89+1 (2)
91=89+2 (2)
92=89+3 (2)
93=89+3+1 (3)
94=89+5 (2)
95=89+5+1 (3)
96=89+5+2 (3)
97=89+8 (2)
98=89+8+1 (3)
99=89+8+2 (3)
100=89+8+3 (3)
101=89+8+3+1 (4)
102=89+13 (2)
103=89+13+1 (3)
104=89+13+2 (3)
105=89+13+3 (3)
106=89+13+3+1 (4)
107=89+13+5 (3)
108=89+13+5+1 (4)
109=89+13+5+2 (4)
110=89+21 (2)
111=89+21+1 (3)
112=89+21+2 (3)
113=89+21+3 (3)
114=89+21+3+1 (4)
115=89+21+5 (3)
116=89+21+5+1 (4)
117=89+21+5+2 (4)
118=89+21+8 (3)
119=89+21+8+1 (4)
120=89+21+8+2 (4)
121=89+21+8+3 (4)
122=89+21+8+3+1 (5)
123=89+34 (2)
124=89+34+1 (3)
125=89+34+2 (3)
126=89+34+3 (3)
127=89+34+3+1 (4)
128=89+34+5 (3)
129=89+34+5+1 (4)
130=89+34+5+2 (4)
131=89+34+8 (3)
132=89+34+8+1 (4)
133=89+34+8+2 (4)
134=89+34+8+3 (4)
135=89+34+8+3+1 (5)
136=89+34+13 (3)
137=89+34+13+1 (4)
138=89+34+13+2 (4)
139=89+34+13+3 (4)
140=89+34+13+3+1 (5)
141=89+34+13+5 (4)
142=89+34+13+5+1 (5)
143=89+34+13+5+2 (5)
144=144 (1)
145=144+1 (2)
146=144+2 (2)
147=144+3 (2)
148=144+3+1 (3)
149=144+5 (2)
150=144+5+1 (3)
151=144+5+2 (3)
152=144+8 (2)
153=144+8+1 (3)
154=144+8+2 (3)
155=144+8+3 (3)
156=144+8+3+1 (4)
157=144+13 (2)
158=144+13+1 (3)
159=144+13+2 (3)
160=144+13+3 (3)
161=144+13+3+1 (4)
162=144+13+5 (3)
163=144+13+5+1 (4)
164=144+13+5+2 (4)
165=144+21 (2)
166=144+21+1 (3)
167=144+21+2 (3)
168=144+21+3 (3)
169=144+21+3+1 (4)
170=144+21+5 (3)
171=144+21+5+1 (4)
172=144+21+5+2 (4)
173=144+21+8 (3)
174=144+21+8+1 (4)
175=144+21+8+2 (4)
176=144+21+8+3 (4)
177=144+21+8+3+1 (5)
178=144+34 (2)
179=144+34+1 (3)
180=144+34+2 (3)
181=144+34+3 (3)
182=144+34+3+1 (4)
183=144+34+5 (3)
184=144+34+5+1 (4)
185=144+34+5+2 (4)
186=144+34+8 (3)
187=144+34+8+1 (4)
188=144+34+8+2 (4)
189=144+34+8+3 (4)
190=144+34+8+3+1 (5)
191=144+34+13 (3)
192=144+34+13+1 (4)
193=144+34+13+2 (4)
194=144+34+13+3 (4)
195=144+34+13+3+1 (5)
196=144+34+13+5 (4)
197=144+34+13+5+1 (5)
198=144+34+13+5+2 (5)
199=144+55 (2)
200=144+55+1 (3)
201=144+55+2 (3)
202=144+55+3 (3)
203=144+55+3+1 (4)
204=144+55+5 (3)
205=144+55+5+1 (4)
206=144+55+5+2 (4)
207=144+55+8 (3)
208=144+55+8+1 (4)
209=144+55+8+2 (4)
210=144+55+8+3 (4)
211=144+55+8+3+1 (5)
212=144+55+13 (3)
213=144+55+13+1 (4)
214=144+55+13+2 (4)
215=144+55+13+3 (4)
216=144+55+13+3+1 (5)
217=144+55+13+5 (4)
218=144+55+13+5+1 (5)
219=144+55+13+5+2 (5)
220=144+55+21 (3)
221=144+55+21+1 (4)
222=144+55+21+2 (4)
223=144+55+21+3 (4)
224=144+55+21+3+1 (5)
225=144+55+21+5 (4)
226=144+55+21+5+1 (5)
227=144+55+21+5+2 (5)
228=144+55+21+8 (4)
229=144+55+21+8+1 (5)
230=144+55+21+8+2 (5)
231=144+55+21+8+3 (5)
232=144+55+21+8+3+1 (6)
233=233 (1)
234=233+1 (2)
235=233+2 (2)
236=233+3 (2)
237=233+3+1 (3)
238=233+5 (2)
239=233+5+1 (3)
240=233+5+2 (3)
241=233+8 (2)
242=233+8+1 (3)
243=233+8+2 (3)
244=233+8+3 (3)
245=233+8+3+1 (4)
246=233+13 (2)
247=233+13+1 (3)
248=233+13+2 (3)
249=233+13+3 (3)
250=233+13+3+1 (4)
251=233+13+5 (3)
252=233+13+5+1 (4)
253=233+13+5+2 (4)
254=233+21 (2)
255=233+21+1 (3)
256=233+21+2 (3)
257=233+21+3 (3)
258=233+21+3+1 (4)
259=233+21+5 (3)
260=233+21+5+1 (4)
261=233+21+5+2 (4)
262=233+21+8 (3)
263=233+21+8+1 (4)
264=233+21+8+2 (4)
265=233+21+8+3 (4)
266=233+21+8+3+1 (5)
267=233+34 (2)
268=233+34+1 (3)
269=233+34+2 (3)
270=233+34+3 (3)
271=233+34+3+1 (4)
272=233+34+5 (3)
273=233+34+5+1 (4)
274=233+34+5+2 (4)
275=233+34+8 (3)
276=233+34+8+1 (4)
277=233+34+8+2 (4)
278=233+34+8+3 (4)
279=233+34+8+3+1 (5)
280=233+34+13 (3)
281=233+34+13+1 (4)
282=233+34+13+2 (4)
283=233+34+13+3 (4)
284=233+34+13+3+1 (5)
285=233+34+13+5 (4)
286=233+34+13+5+1 (5)
287=233+34+13+5+2 (5)
288=233+55 (2)
289=233+55+1 (3)
290=233+55+2 (3)
291=233+55+3 (3)
292=233+55+3+1 (4)
293=233+55+5 (3)
294=233+55+5+1 (4)
295=233+55+5+2 (4)
296=233+55+8 (3)
297=233+55+8+1 (4)
298=233+55+8+2 (4)
299=233+55+8+3 (4)
300=233+55+8+3+1 (5)
301=233+55+13 (3)
302=233+55+13+1 (4)
303=233+55+13+2 (4)
304=233+55+13+3 (4)
305=233+55+13+3+1 (5)
306=233+55+13+5 (4)
307=233+55+13+5+1 (5)
308=233+55+13+5+2 (5)
309=233+55+21 (3)
310=233+55+21+1 (4)
311=233+55+21+2 (4)
312=233+55+21+3 (4)
313=233+55+21+3+1 (5)
314=233+55+21+5 (4)
315=233+55+21+5+1 (5)
316=233+55+21+5+2 (5)
317=233+55+21+8 (4)
318=233+55+21+8+1 (5)
319=233+55+21+8+2 (5)
320=233+55+21+8+3 (5)
321=233+55+21+8+3+1 (6)
322=233+89 (2)
323=233+89+1 (3)
324=233+89+2 (3)
325=233+89+3 (3)
326=233+89+3+1 (4)
327=233+89+5 (3)
328=233+89+5+1 (4)
329=233+89+5+2 (4)
330=233+89+8 (3)
331=233+89+8+1 (4)
332=233+89+8+2 (4)
333=233+89+8+3 (4)
334=233+89+8+3+1 (5)
335=233+89+13 (3)
336=233+89+13+1 (4)
337=233+89+13+2 (4)
338=233+89+13+3 (4)
339=233+89+13+3+1 (5)
340=233+89+13+5 (4)
341=233+89+13+5+1 (5)
342=233+89+13+5+2 (5)
343=233+89+21 (3)
344=233+89+21+1 (4)
345=233+89+21+2 (4)
346=233+89+21+3 (4)
347=233+89+21+3+1 (5)
348=233+89+21+5 (4)
349=233+89+21+5+1 (5)
350=233+89+21+5+2 (5)
351=233+89+21+8 (4)
352=233+89+21+8+1 (5)
353=233+89+21+8+2 (5)
354=233+89+21+8+3 (5)
355=233+89+21+8+3+1 (6)
356=233+89+34 (3)
357=233+89+34+1 (4)
358=233+89+34+2 (4)
359=233+89+34+3 (4)
360=233+89+34+3+1 (5)
361=233+89+34+5 (4)
362=233+89+34+5+1 (5)
363=233+89+34+5+2 (5)
364=233+89+34+8 (4)
365=233+89+34+8+1 (5)
366=233+89+34+8+2 (5)
367=233+89+34+8+3 (5)
368=233+89+34+8+3+1 (6)
369=233+89+34+13 (4)
370=233+89+34+13+1 (5)
371=233+89+34+13+2 (5)
372=233+89+34+13+3 (5)
373=233+89+34+13+3+1 (6)
374=233+89+34+13+5 (5)
375=233+89+34+13+5+1 (6)
376=233+89+34+13+5+2 (6)
377=377 (1)
378=377+1 (2)
379=377+2 (2)
380=377+3 (2)
381=377+3+1 (3)
382=377+5 (2)
383=377+5+1 (3)
384=377+5+2 (3)
385=377+8 (2)
386=377+8+1 (3)
387=377+8+2 (3)
388=377+8+3 (3)
389=377+8+3+1 (4)
390=377+13 (2)
391=377+13+1 (3)
392=377+13+2 (3)
393=377+13+3 (3)
394=377+13+3+1 (4)
395=377+13+5 (3)
396=377+13+5+1 (4)
397=377+13+5+2 (4)
398=377+21 (2)
399=377+21+1 (3)
400=377+21+2 (3)
401=377+21+3 (3)
402=377+21+3+1 (4)
403=377+21+5 (3)
404=377+21+5+1 (4)
405=377+21+5+2 (4)
406=377+21+8 (3)
407=377+21+8+1 (4)
408=377+21+8+2 (4)
409=377+21+8+3 (4)
410=377+21+8+3+1 (5)
411=377+34 (2)
412=377+34+1 (3)
413=377+34+2 (3)
414=377+34+3 (3)
415=377+34+3+1 (4)
416=377+34+5 (3)
417=377+34+5+1 (4)
418=377+34+5+2 (4)
419=377+34+8 (3)
420=377+34+8+1 (4)
421=377+34+8+2 (4)
422=377+34+8+3 (4)
423=377+34+8+3+1 (5)
424=377+34+13 (3)
425=377+34+13+1 (4)
426=377+34+13+2 (4)
427=377+34+13+3 (4)
428=377+34+13+3+1 (5)
429=377+34+13+5 (4)
430=377+34+13+5+1 (5)
431=377+34+13+5+2 (5)
432=377+55 (2)
433=377+55+1 (3)
434=377+55+2 (3)
435=377+55+3 (3)
436=377+55+3+1 (4)
437=377+55+5 (3)
438=377+55+5+1 (4)
439=377+55+5+2 (4)
440=377+55+8 (3)
441=377+55+8+1 (4)
442=377+55+8+2 (4)
443=377+55+8+3 (4)
444=377+55+8+3+1 (5)
445=377+55+13 (3)
446=377+55+13+1 (4)
447=377+55+13+2 (4)
448=377+55+13+3 (4)
449=377+55+13+3+1 (5)
450=377+55+13+5 (4)
451=377+55+13+5+1 (5)
452=377+55+13+5+2 (5)
453=377+55+21 (3)
454=377+55+21+1 (4)
455=377+55+21+2 (4)
456=377+55+21+3 (4)
457=377+55+21+3+1 (5)
458=377+55+21+5 (4)
459=377+55+21+5+1 (5)
460=377+55+21+5+2 (5)
461=377+55+21+8 (4)
462=377+55+21+8+1 (5)
463=377+55+21+8+2 (5)
464=377+55+21+8+3 (5)
465=377+55+21+8+3+1 (6)
466=377+89 (2)
467=377+89+1 (3)
468=377+89+2 (3)
469=377+89+3 (3)
470=377+89+3+1 (4)
471=377+89+5 (3)
472=377+89+5+1 (4)
473=377+89+5+2 (4)
474=377+89+8 (3)
475=377+89+8+1 (4)
476=377+89+8+2 (4)
477=377+89+8+3 (4)
478=377+89+8+3+1 (5)
479=377+89+13 (3)
480=377+89+13+1 (4)
481=377+89+13+2 (4)
482=377+89+13+3 (4)
483=377+89+13+3+1 (5)
484=377+89+13+5 (4)
485=377+89+13+5+1 (5)
486=377+89+13+5+2 (5)
487=377+89+21 (3)
488=377+89+21+1 (4)
489=377+89+21+2 (4)
490=377+89+21+3 (4)
491=377+89+21+3+1 (5)
492=377+89+21+5 (4)
493=377+89+21+5+1 (5)
494=377+89+21+5+2 (5)
495=377+89+21+8 (4)
496=377+89+21+8+1 (5)
497=377+89+21+8+2 (5)
498=377+89+21+8+3 (5)
499=377+89+21+8+3+1 (6)
500=377+89+34 (3)
501=377+89+34+1 (4)
502=377+89+34+2 (4)
503=377+89+34+3 (4)
504=377+89+34+3+1 (5)
505=377+89+34+5 (4)
506=377+89+34+5+1 (5)
507=377+89+34+5+2 (5)
508=377+89+34+8 (4)
509=377+89+34+8+1 (5)
510=377+89+34+8+2 (5)
511=377+89+34+8+3 (5)
512=377+89+34+8+3+1 (6)
513=377+89+34+13 (4)
514=377+89+34+13+1 (5)
515=377+89+34+13+2 (5)
516=377+89+34+13+3 (5)
517=377+89+34+13+3+1 (6)
518=377+89+34+13+5 (5)
519=377+89+34+13+5+1 (6)
520=377+89+34+13+5+2 (6)
521=377+144 (2)
522=377+144+1 (3)
523=377+144+2 (3)
524=377+144+3 (3)
525=377+144+3+1 (4)
526=377+144+5 (3)
527=377+144+5+1 (4)
528=377+144+5+2 (4)
529=377+144+8 (3)
530=377+144+8+1 (4)
531=377+144+8+2 (4)
532=377+144+8+3 (4)
533=377+144+8+3+1 (5)
534=377+144+13 (3)
535=377+144+13+1 (4)
536=377+144+13+2 (4)
537=377+144+13+3 (4)
538=377+144+13+3+1 (5)
539=377+144+13+5 (4)
540=377+144+13+5+1 (5)
541=377+144+13+5+2 (5)
542=377+144+21 (3)
543=377+144+21+1 (4)
544=377+144+21+2 (4)
545=377+144+21+3 (4)
546=377+144+21+3+1 (5)
547=377+144+21+5 (4)
548=377+144+21+5+1 (5)
549=377+144+21+5+2 (5)
550=377+144+21+8 (4)
551=377+144+21+8+1 (5)
552=377+144+21+8+2 (5)
553=377+144+21+8+3 (5)
554=377+144+21+8+3+1 (6)
555=377+144+34 (3)
556=377+144+34+1 (4)
557=377+144+34+2 (4)
558=377+144+34+3 (4)
559=377+144+34+3+1 (5)
560=377+144+34+5 (4)
561=377+144+34+5+1 (5)
562=377+144+34+5+2 (5)
563=377+144+34+8 (4)
564=377+144+34+8+1 (5)
565=377+144+34+8+2 (5)
566=377+144+34+8+3 (5)
567=377+144+34+8+3+1 (6)
568=377+144+34+13 (4)
569=377+144+34+13+1 (5)
570=377+144+34+13+2 (5)
571=377+144+34+13+3 (5)
572=377+144+34+13+3+1 (6)
573=377+144+34+13+5 (5)
574=377+144+34+13+5+1 (6)
575=377+144+34+13+5+2 (6)
576=377+144+55 (3)
577=377+144+55+1 (4)
578=377+144+55+2 (4)
579=377+144+55+3 (4)
580=377+144+55+3+1 (5)
581=377+144+55+5 (4)
582=377+144+55+5+1 (5)
583=377+144+55+5+2 (5)
584=377+144+55+8 (4)
585=377+144+55+8+1 (5)
586=377+144+55+8+2 (5)
587=377+144+55+8+3 (5)
588=377+144+55+8+3+1 (6)
589=377+144+55+13 (4)
590=377+144+55+13+1 (5)
591=377+144+55+13+2 (5)
592=377+144+55+13+3 (5)
593=377+144+55+13+3+1 (6)
594=377+144+55+13+5 (5)
595=377+144+55+13+5+1 (6)
596=377+144+55+13+5+2 (6)
597=377+144+55+21 (4)
598=377+144+55+21+1 (5)
599=377+144+55+21+2 (5)
600=377+144+55+21+3 (5)
601=377+144+55+21+3+1 (6)
602=377+144+55+21+5 (5)
603=377+144+55+21+5+1 (6)
604=377+144+55+21+5+2 (6)
605=377+144+55+21+8 (5)
606=377+144+55+21+8+1 (6)
607=377+144+55+21+8+2 (6)
608=377+144+55+21+8+3 (6)
609=377+144+55+21+8+3+1 (7)
610=610 (1)
611=610+1 (2)
612=610+2 (2)
613=610+3 (2)
614=610+3+1 (3)
615=610+5 (2)
616=610+5+1 (3)
617=610+5+2 (3)
618=610+8 (2)
619=610+8+1 (3)
620=610+8+2 (3)
621=610+8+3 (3)
622=610+8+3+1 (4)
623=610+13 (2)
624=610+13+1 (3)
625=610+13+2 (3)
626=610+13+3 (3)
627=610+13+3+1 (4)
628=610+13+5 (3)
629=610+13+5+1 (4)
630=610+13+5+2 (4)
631=610+21 (2)
632=610+21+1 (3)
633=610+21+2 (3)
634=610+21+3 (3)
635=610+21+3+1 (4)
636=610+21+5 (3)
637=610+21+5+1 (4)
638=610+21+5+2 (4)
639=610+21+8 (3)
640=610+21+8+1 (4)
641=610+21+8+2 (4)
642=610+21+8+3 (4)
643=610+21+8+3+1 (5)
644=610+34 (2)
645=610+34+1 (3)
646=610+34+2 (3)
647=610+34+3 (3)
648=610+34+3+1 (4)
649=610+34+5 (3)
650=610+34+5+1 (4)
651=610+34+5+2 (4)
652=610+34+8 (3)
653=610+34+8+1 (4)
654=610+34+8+2 (4)
655=610+34+8+3 (4)
656=610+34+8+3+1 (5)
657=610+34+13 (3)
658=610+34+13+1 (4)
659=610+34+13+2 (4)
660=610+34+13+3 (4)
661=610+34+13+3+1 (5)
662=610+34+13+5 (4)
663=610+34+13+5+1 (5)
664=610+34+13+5+2 (5)
665=610+55 (2)
666=610+55+1 (3)
667=610+55+2 (3)
668=610+55+3 (3)
669=610+55+3+1 (4)
670=610+55+5 (3)
671=610+55+5+1 (4)
672=610+55+5+2 (4)
673=610+55+8 (3)
674=610+55+8+1 (4)
675=610+55+8+2 (4)
676=610+55+8+3 (4)
677=610+55+8+3+1 (5)
678=610+55+13 (3)
679=610+55+13+1 (4)
680=610+55+13+2 (4)
681=610+55+13+3 (4)
682=610+55+13+3+1 (5)
683=610+55+13+5 (4)
684=610+55+13+5+1 (5)
685=610+55+13+5+2 (5)
686=610+55+21 (3)
687=610+55+21+1 (4)
688=610+55+21+2 (4)
689=610+55+21+3 (4)
690=610+55+21+3+1 (5)
691=610+55+21+5 (4)
692=610+55+21+5+1 (5)
693=610+55+21+5+2 (5)
694=610+55+21+8 (4)
695=610+55+21+8+1 (5)
696=610+55+21+8+2 (5)
697=610+55+21+8+3 (5)
698=610+55+21+8+3+1 (6)
699=610+89 (2)
700=610+89+1 (3)
701=610+89+2 (3)
702=610+89+3 (3)
703=610+89+3+1 (4)
704=610+89+5 (3)
705=610+89+5+1 (4)
706=610+89+5+2 (4)
707=610+89+8 (3)
708=610+89+8+1 (4)
709=610+89+8+2 (4)
710=610+89+8+3 (4)
711=610+89+8+3+1 (5)
712=610+89+13 (3)
713=610+89+13+1 (4)
714=610+89+13+2 (4)
715=610+89+13+3 (4)
716=610+89+13+3+1 (5)
717=610+89+13+5 (4)
718=610+89+13+5+1 (5)
719=610+89+13+5+2 (5)
720=610+89+21 (3)
721=610+89+21+1 (4)
722=610+89+21+2 (4)
723=610+89+21+3 (4)
724=610+89+21+3+1 (5)
725=610+89+21+5 (4)
726=610+89+21+5+1 (5)
727=610+89+21+5+2 (5)
728=610+89+21+8 (4)
729=610+89+21+8+1 (5)
730=610+89+21+8+2 (5)
731=610+89+21+8+3 (5)
732=610+89+21+8+3+1 (6)
733=610+89+34 (3)
734=610+89+34+1 (4)
735=610+89+34+2 (4)
736=610+89+34+3 (4)
737=610+89+34+3+1 (5)
738=610+89+34+5 (4)
739=610+89+34+5+1 (5)
740=610+89+34+5+2 (5)
741=610+89+34+8 (4)
742=610+89+34+8+1 (5)
743=610+89+34+8+2 (5)
744=610+89+34+8+3 (5)
745=610+89+34+8+3+1 (6)
746=610+89+34+13 (4)
747=610+89+34+13+1 (5)
748=610+89+34+13+2 (5)
749=610+89+34+13+3 (5)
750=610+89+34+13+3+1 (6)
751=610+89+34+13+5 (5)
752=610+89+34+13+5+1 (6)
753=610+89+34+13+5+2 (6)
754=610+144 (2)
755=610+144+1 (3)
756=610+144+2 (3)
757=610+144+3 (3)
758=610+144+3+1 (4)
759=610+144+5 (3)
760=610+144+5+1 (4)
761=610+144+5+2 (4)
762=610+144+8 (3)
763=610+144+8+1 (4)
764=610+144+8+2 (4)
765=610+144+8+3 (4)
766=610+144+8+3+1 (5)
767=610+144+13 (3)
768=610+144+13+1 (4)
769=610+144+13+2 (4)
770=610+144+13+3 (4)
771=610+144+13+3+1 (5)
772=610+144+13+5 (4)
773=610+144+13+5+1 (5)
774=610+144+13+5+2 (5)
775=610+144+21 (3)
776=610+144+21+1 (4)
777=610+144+21+2 (4)
778=610+144+21+3 (4)
779=610+144+21+3+1 (5)
780=610+144+21+5 (4)
781=610+144+21+5+1 (5)
782=610+144+21+5+2 (5)
783=610+144+21+8 (4)
784=610+144+21+8+1 (5)
785=610+144+21+8+2 (5)
786=610+144+21+8+3 (5)
787=610+144+21+8+3+1 (6)
788=610+144+34 (3)
789=610+144+34+1 (4)
790=610+144+34+2 (4)
791=610+144+34+3 (4)
792=610+144+34+3+1 (5)
793=610+144+34+5 (4)
794=610+144+34+5+1 (5)
795=610+144+34+5+2 (5)
796=610+144+34+8 (4)
797=610+144+34+8+1 (5)
798=610+144+34+8+2 (5)
799=610+144+34+8+3 (5)
800=610+144+34+8+3+1 (6)
801=610+144+34+13 (4)
802=610+144+34+13+1 (5)
803=610+144+34+13+2 (5)
804=610+144+34+13+3 (5)
805=610+144+34+13+3+1 (6)
806=610+144+34+13+5 (5)
807=610+144+34+13+5+1 (6)
808=610+144+34+13+5+2 (6)
809=610+144+55 (3)
810=610+144+55+1 (4)
811=610+144+55+2 (4)
812=610+144+55+3 (4)
813=610+144+55+3+1 (5)
814=610+144+55+5 (4)
815=610+144+55+5+1 (5)
816=610+144+55+5+2 (5)
817=610+144+55+8 (4)
818=610+144+55+8+1 (5)
819=610+144+55+8+2 (5)
820=610+144+55+8+3 (5)
821=610+144+55+8+3+1 (6)
822=610+144+55+13 (4)
823=610+144+55+13+1 (5)
824=610+144+55+13+2 (5)
825=610+144+55+13+3 (5)
826=610+144+55+13+3+1 (6)
827=610+144+55+13+5 (5)
828=610+144+55+13+5+1 (6)
829=610+144+55+13+5+2 (6)
830=610+144+55+21 (4)
831=610+144+55+21+1 (5)
832=610+144+55+21+2 (5)
833=610+144+55+21+3 (5)
834=610+144+55+21+3+1 (6)
835=610+144+55+21+5 (5)
836=610+144+55+21+5+1 (6)
837=610+144+55+21+5+2 (6)
838=610+144+55+21+8 (5)
839=610+144+55+21+8+1 (6)
840=610+144+55+21+8+2 (6)
841=610+144+55+21+8+3 (6)
842=610+144+55+21+8+3+1 (7)
843=610+233 (2)
844=610+233+1 (3)
845=610+233+2 (3)
846=610+233+3 (3)
847=610+233+3+1 (4)
848=610+233+5 (3)
849=610+233+5+1 (4)
850=610+233+5+2 (4)
851=610+233+8 (3)
852=610+233+8+1 (4)
853=610+233+8+2 (4)
854=610+233+8+3 (4)
855=610+233+8+3+1 (5)
856=610+233+13 (3)
857=610+233+13+1 (4)
858=610+233+13+2 (4)
859=610+233+13+3 (4)
860=610+233+13+3+1 (5)
861=610+233+13+5 (4)
862=610+233+13+5+1 (5)
863=610+233+13+5+2 (5)
864=610+233+21 (3)
865=610+233+21+1 (4)
866=610+233+21+2 (4)
867=610+233+21+3 (4)
868=610+233+21+3+1 (5)
869=610+233+21+5 (4)
870=610+233+21+5+1 (5)
871=610+233+21+5+2 (5)
872=610+233+21+8 (4)
873=610+233+21+8+1 (5)
874=610+233+21+8+2 (5)
875=610+233+21+8+3 (5)
876=610+233+21+8+3+1 (6)
877=610+233+34 (3)
878=610+233+34+1 (4)
879=610+233+34+2 (4)
880=610+233+34+3 (4)
881=610+233+34+3+1 (5)
882=610+233+34+5 (4)
883=610+233+34+5+1 (5)
884=610+233+34+5+2 (5)
885=610+233+34+8 (4)
886=610+233+34+8+1 (5)
887=610+233+34+8+2 (5)
888=610+233+34+8+3 (5)
889=610+233+34+8+3+1 (6)
890=610+233+34+13 (4)
891=610+233+34+13+1 (5)
892=610+233+34+13+2 (5)
893=610+233+34+13+3 (5)
894=610+233+34+13+3+1 (6)
895=610+233+34+13+5 (5)
896=610+233+34+13+5+1 (6)
897=610+233+34+13+5+2 (6)
898=610+233+55 (3)
899=610+233+55+1 (4)
900=610+233+55+2 (4)
901=610+233+55+3 (4)
902=610+233+55+3+1 (5)
903=610+233+55+5 (4)
904=610+233+55+5+1 (5)
905=610+233+55+5+2 (5)
906=610+233+55+8 (4)
907=610+233+55+8+1 (5)
908=610+233+55+8+2 (5)
909=610+233+55+8+3 (5)
910=610+233+55+8+3+1 (6)
911=610+233+55+13 (4)
912=610+233+55+13+1 (5)
913=610+233+55+13+2 (5)
914=610+233+55+13+3 (5)
915=610+233+55+13+3+1 (6)
916=610+233+55+13+5 (5)
917=610+233+55+13+5+1 (6)
918=610+233+55+13+5+2 (6)
919=610+233+55+21 (4)
920=610+233+55+21+1 (5)
921=610+233+55+21+2 (5)
922=610+233+55+21+3 (5)
923=610+233+55+21+3+1 (6)
924=610+233+55+21+5 (5)
925=610+233+55+21+5+1 (6)
926=610+233+55+21+5+2 (6)
927=610+233+55+21+8 (5)
928=610+233+55+21+8+1 (6)
929=610+233+55+21+8+2 (6)
930=610+233+55+21+8+3 (6)
931=610+233+55+21+8+3+1 (7)
932=610+233+89 (3)
933=610+233+89+1 (4)
934=610+233+89+2 (4)
935=610+233+89+3 (4)
936=610+233+89+3+1 (5)
937=610+233+89+5 (4)
938=610+233+89+5+1 (5)
939=610+233+89+5+2 (5)
940=610+233+89+8 (4)
941=610+233+89+8+1 (5)
942=610+233+89+8+2 (5)
943=610+233+89+8+3 (5)
944=610+233+89+8+3+1 (6)
945=610+233+89+13 (4)
946=610+233+89+13+1 (5)
947=610+233+89+13+2 (5)
948=610+233+89+13+3 (5)
949=610+233+89+13+3+1 (6)
950=610+233+89+13+5 (5)
951=610+233+89+13+5+1 (6)
952=610+233+89+13+5+2 (6)
953=610+233+89+21 (4)
954=610+233+89+21+1 (5)
955=610+233+89+21+2 (5)
956=610+233+89+21+3 (5)
957=610+233+89+21+3+1 (6)
958=610+233+89+21+5 (5)
959=610+233+89+21+5+1 (6)
960=610+233+89+21+5+2 (6)
961=610+233+89+21+8 (5)
962=610+233+89+21+8+1 (6)
963=610+233+89+21+8+2 (6)
964=610+233+89+21+8+3 (6)
965=610+233+89+21+8+3+1 (7)
966=610+233+89+34 (4)
967=610+233+89+34+1 (5)
968=610+233+89+34+2 (5)
969=610+233+89+34+3 (5)
970=610+233+89+34+3+1 (6)
971=610+233+89+34+5 (5)
972=610+233+89+34+5+1 (6)
973=610+233+89+34+5+2 (6)
974=610+233+89+34+8 (5)
975=610+233+89+34+8+1 (6)
976=610+233+89+34+8+2 (6)
977=610+233+89+34+8+3 (6)
978=610+233+89+34+8+3+1 (7)
979=610+233+89+34+13 (5)
980=610+233+89+34+13+1 (6)
981=610+233+89+34+13+2 (6)
982=610+233+89+34+13+3 (6)
983=610+233+89+34+13+3+1 (7)
984=610+233+89+34+13+5 (6)
985=610+233+89+34+13+5+1 (7)
986=610+233+89+34+13+5+2 (7)
987=987 (1)
…
(the following are some large numbers, showing the position, not value, of the 2-fib numbers summing to them.
1100087778366101931=88 (1)
1100087778366101930=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5+3 (43)
1100087778366101929=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5+2 (43)
1100087778366101928=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5 (42)

Here’s the MUMPS code that generated it

s N=2 k F s F(1)=1 f A=2:1 q:F(A-1)>9999999999999999999  f I=A-N:1:A-1 s F(A)=$g(F(A))+$g(F(I)) ;build n-fib sequence
s A9=$o(F(""),-1) f B=1:1:987 w !,B s C=B,T=0 x "f A=A9:-1:1 i F(A)'>C w $s('T:""="",1:""+""),F(A) s C=C-F(A),T=T+1 q:'C" w " (",T,")" i C w !,B," no solution!" q  ;find and list sums

PS: maybe this thread should be broken into a couple of separate threads and put in the math forum? It’s become a bit unwatercoolerish :hyper:

Turtle · March 31, 2007

I feel tantalizingly close to a proof of the “the digbog conjecture”, following the lines of the question “is there a Fibonacci number F(n) such that a number x, F(n-1)<x<F(n) can’t be expressed as a sum of non-repeating Fibonacci numbers?” – though in the realm of math intuition “tantalizingly close” sometimes means “not even on the right track”. ;)

So in your expressions, n is the ordinal? That is to say, if I denote the 'fifth Fibonacci number' then n=5?

A few observations:
The conjecture is false for some generalized Fibonacci sequences, such as ”non-minimal” n-fib sequences (that is, any that don’t start {0,1,1,2} or similarly), such as {1,3,4,7,11 …}

What does the n here in 'n-fib' take its value from? Also, 0 (zero) is not in the Fibonacci set so its unclear to me where you get the set {0,1,1,2}?

The conjecture appears true for any “minimal” n-fib sequence, such as the 3-fib sequence {0,0,1,1,2,4,7,11 …} or the 9-fib sequence {1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144,16272,32512,64960,129792}

I have the same confusion on this point as above, i.e. what is the n in n-fib, and why is 0 in the set?

It appears to always be possible to find the terms that sum to a number by “working backward”, choosing the greatest fib number not greater than the remainder of the number minus the previously chosen numbers.

I think this describes the by-hand procedure I have been using. Beginning with a non-Fibonacci number n, I subtract the largest Fibonacci number that is smaller than n, then from that resulting difference I again subtract the largest Fibonacci number that is smaller, and so on 'til I fill the sum.

No “intelligence” is required to find the terms summing to any number.

and a conjecture of my own
No number less than F(n)-1 will be the sum of more terms than the number of terms summing to F(n)-1

I think I agree; like 88, yes?

Just to demonstrate the unimpressive fact that I have a computer and a programming system that allows me to crank out numbers with ease (and for the viewing convenience of anyone who doesn’t), here’s a list of 1 through 987 as sums of 2-fib numbers, with a bit of extra data at the end:

What impresses me is your solid grasp of mathematics & your willingness to contribute your time & effort. Also your wry sense of humor. Fish on!!! :doh:

1=1 (1)
2=2 (1)
3=3 (1)
4=3+1 (2)
5=5 (1)
6=5+1 (2)
7=5+2 (2)
8=8 (1)
9=8+1 (2)
10=8+2 (2)
11=8+3 (2)
12=8+3+1 (3)
13=13 (1)
14=13+1 (2)
15=13+2 (2)
16=13+3 (2)
17=13+3+1 (3)
18=13+5 (2)
19=13+5+1 (3)
20=13+5+2 (3)
21=21 (1)
22=21+1 (2)
23=21+2 (2)
24=21+3 (2)
25=21+3+1 (3)
26=21+5 (2)
27=21+5+1 (3)
28=21+5+2 (3)
29=21+8 (2)
30=21+8+1 (3)
31=21+8+2 (3)
32=21+8+3 (3)
33=21+8+3+1 (4)
34=34 (1)
35=34+1 (2)
36=34+2 (2)
37=34+3 (2)
38=34+3+1 (3)
39=34+5 (2)
40=34+5+1 (3)
41=34+5+2 (3)
42=34+8 (2)
43=34+8+1 (3)
44=34+8+2 (3)
45=34+8+3 (3)
46=34+8+3+1 (4)
47=34+13 (2)
48=34+13+1 (3)
49=34+13+2 (3)
50=34+13+3 (3)
51=34+13+3+1 (4)
52=34+13+5 (3)
53=34+13+5+1 (4)
54=34+13+5+2 (4)
55=55 (1)
56=55+1 (2)
57=55+2 (2)
58=55+3 (2)
59=55+3+1 (3)
60=55+5 (2)
61=55+5+1 (3)
62=55+5+2 (3)
63=55+8 (2)
64=55+8+1 (3)
65=55+8+2 (3)
66=55+8+3 (3)
67=55+8+3+1 (4)
68=55+13 (2)
69=55+13+1 (3)
70=55+13+2 (3)
71=55+13+3 (3)
72=55+13+3+1 (4)
73=55+13+5 (3)
74=55+13+5+1 (4)
75=55+13+5+2 (4)
76=55+21 (2)
77=55+21+1 (3)
78=55+21+2 (3)
79=55+21+3 (3)
80=55+21+3+1 (4)
81=55+21+5 (3)
82=55+21+5+1 (4)
83=55+21+5+2 (4)
84=55+21+8 (3)
85=55+21+8+1 (4)
86=55+21+8+2 (4)
87=55+21+8+3 (4)
88=55+21+8+3+1 (5)
89=89 (1)
90=89+1 (2)
91=89+2 (2)
92=89+3 (2)
93=89+3+1 (3)
94=89+5 (2)
95=89+5+1 (3)
96=89+5+2 (3)
97=89+8 (2)
98=89+8+1 (3)
99=89+8+2 (3)
100=89+8+3 (3)
101=89+8+3+1 (4)
102=89+13 (2)
103=89+13+1 (3)
104=89+13+2 (3)
105=89+13+3 (3)
106=89+13+3+1 (4)
107=89+13+5 (3)
108=89+13+5+1 (4)
109=89+13+5+2 (4)
110=89+21 (2)
111=89+21+1 (3)
112=89+21+2 (3)
113=89+21+3 (3)
114=89+21+3+1 (4)
115=89+21+5 (3)
116=89+21+5+1 (4)
117=89+21+5+2 (4)
118=89+21+8 (3)
119=89+21+8+1 (4)
120=89+21+8+2 (4)
121=89+21+8+3 (4)
122=89+21+8+3+1 (5)
123=89+34 (2)
124=89+34+1 (3)
125=89+34+2 (3)
126=89+34+3 (3)
127=89+34+3+1 (4)
128=89+34+5 (3)
129=89+34+5+1 (4)
130=89+34+5+2 (4)
131=89+34+8 (3)
132=89+34+8+1 (4)
133=89+34+8+2 (4)
134=89+34+8+3 (4)
135=89+34+8+3+1 (5)
136=89+34+13 (3)
137=89+34+13+1 (4)
138=89+34+13+2 (4)
139=89+34+13+3 (4)
140=89+34+13+3+1 (5)
141=89+34+13+5 (4)
142=89+34+13+5+1 (5)
143=89+34+13+5+2 (5)
144=144 (1)
145=144+1 (2)
146=144+2 (2)
147=144+3 (2)
148=144+3+1 (3)
149=144+5 (2)
150=144+5+1 (3)
151=144+5+2 (3)
152=144+8 (2)
153=144+8+1 (3)
154=144+8+2 (3)
155=144+8+3 (3)
156=144+8+3+1 (4)
157=144+13 (2)
158=144+13+1 (3)
159=144+13+2 (3)
160=144+13+3 (3)
161=144+13+3+1 (4)
162=144+13+5 (3)
163=144+13+5+1 (4)
164=144+13+5+2 (4)
165=144+21 (2)
166=144+21+1 (3)
167=144+21+2 (3)
168=144+21+3 (3)
169=144+21+3+1 (4)
170=144+21+5 (3)
171=144+21+5+1 (4)
172=144+21+5+2 (4)
173=144+21+8 (3)
174=144+21+8+1 (4)
175=144+21+8+2 (4)
176=144+21+8+3 (4)
177=144+21+8+3+1 (5)
178=144+34 (2)
179=144+34+1 (3)
180=144+34+2 (3)
181=144+34+3 (3)
182=144+34+3+1 (4)
183=144+34+5 (3)
184=144+34+5+1 (4)
185=144+34+5+2 (4)
186=144+34+8 (3)
187=144+34+8+1 (4)
188=144+34+8+2 (4)
189=144+34+8+3 (4)
190=144+34+8+3+1 (5)
191=144+34+13 (3)
192=144+34+13+1 (4)
193=144+34+13+2 (4)
194=144+34+13+3 (4)
195=144+34+13+3+1 (5)
196=144+34+13+5 (4)
197=144+34+13+5+1 (5)
198=144+34+13+5+2 (5)
199=144+55 (2)
200=144+55+1 (3)
201=144+55+2 (3)
202=144+55+3 (3)
203=144+55+3+1 (4)
204=144+55+5 (3)
205=144+55+5+1 (4)
206=144+55+5+2 (4)
207=144+55+8 (3)
208=144+55+8+1 (4)
209=144+55+8+2 (4)
210=144+55+8+3 (4)
211=144+55+8+3+1 (5)
212=144+55+13 (3)
213=144+55+13+1 (4)
214=144+55+13+2 (4)
215=144+55+13+3 (4)
216=144+55+13+3+1 (5)
217=144+55+13+5 (4)
218=144+55+13+5+1 (5)
219=144+55+13+5+2 (5)
220=144+55+21 (3)
221=144+55+21+1 (4)
222=144+55+21+2 (4)
223=144+55+21+3 (4)
224=144+55+21+3+1 (5)
225=144+55+21+5 (4)
226=144+55+21+5+1 (5)
227=144+55+21+5+2 (5)
228=144+55+21+8 (4)
229=144+55+21+8+1 (5)
230=144+55+21+8+2 (5)
231=144+55+21+8+3 (5)
232=144+55+21+8+3+1 (6)
233=233 (1)
234=233+1 (2)
235=233+2 (2)
236=233+3 (2)
237=233+3+1 (3)
238=233+5 (2)
239=233+5+1 (3)
240=233+5+2 (3)
241=233+8 (2)
242=233+8+1 (3)
243=233+8+2 (3)
244=233+8+3 (3)
245=233+8+3+1 (4)
246=233+13 (2)
247=233+13+1 (3)
248=233+13+2 (3)
249=233+13+3 (3)
250=233+13+3+1 (4)
251=233+13+5 (3)
252=233+13+5+1 (4)
253=233+13+5+2 (4)
254=233+21 (2)
255=233+21+1 (3)
256=233+21+2 (3)
257=233+21+3 (3)
258=233+21+3+1 (4)
259=233+21+5 (3)
260=233+21+5+1 (4)
261=233+21+5+2 (4)
262=233+21+8 (3)
263=233+21+8+1 (4)
264=233+21+8+2 (4)
265=233+21+8+3 (4)
266=233+21+8+3+1 (5)
267=233+34 (2)
268=233+34+1 (3)
269=233+34+2 (3)
270=233+34+3 (3)
271=233+34+3+1 (4)
272=233+34+5 (3)
273=233+34+5+1 (4)
274=233+34+5+2 (4)
275=233+34+8 (3)
276=233+34+8+1 (4)
277=233+34+8+2 (4)
278=233+34+8+3 (4)
279=233+34+8+3+1 (5)
280=233+34+13 (3)
281=233+34+13+1 (4)
282=233+34+13+2 (4)
283=233+34+13+3 (4)
284=233+34+13+3+1 (5)
285=233+34+13+5 (4)
286=233+34+13+5+1 (5)
287=233+34+13+5+2 (5)
288=233+55 (2)
289=233+55+1 (3)
290=233+55+2 (3)
291=233+55+3 (3)
292=233+55+3+1 (4)
293=233+55+5 (3)
294=233+55+5+1 (4)
295=233+55+5+2 (4)
296=233+55+8 (3)
297=233+55+8+1 (4)
298=233+55+8+2 (4)
299=233+55+8+3 (4)
300=233+55+8+3+1 (5)
301=233+55+13 (3)
302=233+55+13+1 (4)
303=233+55+13+2 (4)
304=233+55+13+3 (4)
305=233+55+13+3+1 (5)
306=233+55+13+5 (4)
307=233+55+13+5+1 (5)
308=233+55+13+5+2 (5)
309=233+55+21 (3)
310=233+55+21+1 (4)
311=233+55+21+2 (4)
312=233+55+21+3 (4)
313=233+55+21+3+1 (5)
314=233+55+21+5 (4)
315=233+55+21+5+1 (5)
316=233+55+21+5+2 (5)
317=233+55+21+8 (4)
318=233+55+21+8+1 (5)
319=233+55+21+8+2 (5)
320=233+55+21+8+3 (5)
321=233+55+21+8+3+1 (6)
322=233+89 (2)
323=233+89+1 (3)
324=233+89+2 (3)
325=233+89+3 (3)
326=233+89+3+1 (4)
327=233+89+5 (3)
328=233+89+5+1 (4)
329=233+89+5+2 (4)
330=233+89+8 (3)
331=233+89+8+1 (4)
332=233+89+8+2 (4)
333=233+89+8+3 (4)
334=233+89+8+3+1 (5)
335=233+89+13 (3)
336=233+89+13+1 (4)
337=233+89+13+2 (4)
338=233+89+13+3 (4)
339=233+89+13+3+1 (5)
340=233+89+13+5 (4)
341=233+89+13+5+1 (5)
342=233+89+13+5+2 (5)
343=233+89+21 (3)
344=233+89+21+1 (4)
345=233+89+21+2 (4)
346=233+89+21+3 (4)
347=233+89+21+3+1 (5)
348=233+89+21+5 (4)
349=233+89+21+5+1 (5)
350=233+89+21+5+2 (5)
351=233+89+21+8 (4)
352=233+89+21+8+1 (5)
353=233+89+21+8+2 (5)
354=233+89+21+8+3 (5)
355=233+89+21+8+3+1 (6)
356=233+89+34 (3)
357=233+89+34+1 (4)
358=233+89+34+2 (4)
359=233+89+34+3 (4)
360=233+89+34+3+1 (5)
361=233+89+34+5 (4)
362=233+89+34+5+1 (5)
363=233+89+34+5+2 (5)
364=233+89+34+8 (4)
365=233+89+34+8+1 (5)
366=233+89+34+8+2 (5)
367=233+89+34+8+3 (5)
368=233+89+34+8+3+1 (6)
369=233+89+34+13 (4)
370=233+89+34+13+1 (5)
371=233+89+34+13+2 (5)
372=233+89+34+13+3 (5)
373=233+89+34+13+3+1 (6)
374=233+89+34+13+5 (5)
375=233+89+34+13+5+1 (6)
376=233+89+34+13+5+2 (6)
377=377 (1)
378=377+1 (2)
379=377+2 (2)
380=377+3 (2)
381=377+3+1 (3)
382=377+5 (2)
383=377+5+1 (3)
384=377+5+2 (3)
385=377+8 (2)
386=377+8+1 (3)
387=377+8+2 (3)
388=377+8+3 (3)
389=377+8+3+1 (4)
390=377+13 (2)
391=377+13+1 (3)
392=377+13+2 (3)
393=377+13+3 (3)
394=377+13+3+1 (4)
395=377+13+5 (3)
396=377+13+5+1 (4)
397=377+13+5+2 (4)
398=377+21 (2)
399=377+21+1 (3)
400=377+21+2 (3)
401=377+21+3 (3)
402=377+21+3+1 (4)
403=377+21+5 (3)
404=377+21+5+1 (4)
405=377+21+5+2 (4)
406=377+21+8 (3)
407=377+21+8+1 (4)
408=377+21+8+2 (4)
409=377+21+8+3 (4)
410=377+21+8+3+1 (5)
411=377+34 (2)
412=377+34+1 (3)
413=377+34+2 (3)
414=377+34+3 (3)
415=377+34+3+1 (4)
416=377+34+5 (3)
417=377+34+5+1 (4)
418=377+34+5+2 (4)
419=377+34+8 (3)
420=377+34+8+1 (4)
421=377+34+8+2 (4)
422=377+34+8+3 (4)
423=377+34+8+3+1 (5)
424=377+34+13 (3)
425=377+34+13+1 (4)
426=377+34+13+2 (4)
427=377+34+13+3 (4)
428=377+34+13+3+1 (5)
429=377+34+13+5 (4)
430=377+34+13+5+1 (5)
431=377+34+13+5+2 (5)
432=377+55 (2)
433=377+55+1 (3)
434=377+55+2 (3)
435=377+55+3 (3)
436=377+55+3+1 (4)
437=377+55+5 (3)
438=377+55+5+1 (4)
439=377+55+5+2 (4)
440=377+55+8 (3)
441=377+55+8+1 (4)
442=377+55+8+2 (4)
443=377+55+8+3 (4)
444=377+55+8+3+1 (5)
445=377+55+13 (3)
446=377+55+13+1 (4)
447=377+55+13+2 (4)
448=377+55+13+3 (4)
449=377+55+13+3+1 (5)
450=377+55+13+5 (4)
451=377+55+13+5+1 (5)
452=377+55+13+5+2 (5)
453=377+55+21 (3)
454=377+55+21+1 (4)
455=377+55+21+2 (4)
456=377+55+21+3 (4)
457=377+55+21+3+1 (5)
458=377+55+21+5 (4)
459=377+55+21+5+1 (5)
460=377+55+21+5+2 (5)
461=377+55+21+8 (4)
462=377+55+21+8+1 (5)
463=377+55+21+8+2 (5)
464=377+55+21+8+3 (5)
465=377+55+21+8+3+1 (6)
466=377+89 (2)
467=377+89+1 (3)
468=377+89+2 (3)
469=377+89+3 (3)
470=377+89+3+1 (4)
471=377+89+5 (3)
472=377+89+5+1 (4)
473=377+89+5+2 (4)
474=377+89+8 (3)
475=377+89+8+1 (4)
476=377+89+8+2 (4)
477=377+89+8+3 (4)
478=377+89+8+3+1 (5)
479=377+89+13 (3)
480=377+89+13+1 (4)
481=377+89+13+2 (4)
482=377+89+13+3 (4)
483=377+89+13+3+1 (5)
484=377+89+13+5 (4)
485=377+89+13+5+1 (5)
486=377+89+13+5+2 (5)
487=377+89+21 (3)
488=377+89+21+1 (4)
489=377+89+21+2 (4)
490=377+89+21+3 (4)
491=377+89+21+3+1 (5)
492=377+89+21+5 (4)
493=377+89+21+5+1 (5)
494=377+89+21+5+2 (5)
495=377+89+21+8 (4)
496=377+89+21+8+1 (5)
497=377+89+21+8+2 (5)
498=377+89+21+8+3 (5)
499=377+89+21+8+3+1 (6)
500=377+89+34 (3)
501=377+89+34+1 (4)
502=377+89+34+2 (4)
503=377+89+34+3 (4)
504=377+89+34+3+1 (5)
505=377+89+34+5 (4)
506=377+89+34+5+1 (5)
507=377+89+34+5+2 (5)
508=377+89+34+8 (4)
509=377+89+34+8+1 (5)
510=377+89+34+8+2 (5)
511=377+89+34+8+3 (5)
512=377+89+34+8+3+1 (6)
513=377+89+34+13 (4)
514=377+89+34+13+1 (5)
515=377+89+34+13+2 (5)
516=377+89+34+13+3 (5)
517=377+89+34+13+3+1 (6)
518=377+89+34+13+5 (5)
519=377+89+34+13+5+1 (6)
520=377+89+34+13+5+2 (6)
521=377+144 (2)
522=377+144+1 (3)
523=377+144+2 (3)
524=377+144+3 (3)
525=377+144+3+1 (4)
526=377+144+5 (3)
527=377+144+5+1 (4)
528=377+144+5+2 (4)
529=377+144+8 (3)
530=377+144+8+1 (4)
531=377+144+8+2 (4)
532=377+144+8+3 (4)
533=377+144+8+3+1 (5)
534=377+144+13 (3)
535=377+144+13+1 (4)
536=377+144+13+2 (4)
537=377+144+13+3 (4)
538=377+144+13+3+1 (5)
539=377+144+13+5 (4)
540=377+144+13+5+1 (5)
541=377+144+13+5+2 (5)
542=377+144+21 (3)
543=377+144+21+1 (4)
544=377+144+21+2 (4)
545=377+144+21+3 (4)
546=377+144+21+3+1 (5)
547=377+144+21+5 (4)
548=377+144+21+5+1 (5)
549=377+144+21+5+2 (5)
550=377+144+21+8 (4)
551=377+144+21+8+1 (5)
552=377+144+21+8+2 (5)
553=377+144+21+8+3 (5)
554=377+144+21+8+3+1 (6)
555=377+144+34 (3)
556=377+144+34+1 (4)
557=377+144+34+2 (4)
558=377+144+34+3 (4)
559=377+144+34+3+1 (5)
560=377+144+34+5 (4)
561=377+144+34+5+1 (5)
562=377+144+34+5+2 (5)
563=377+144+34+8 (4)
564=377+144+34+8+1 (5)
565=377+144+34+8+2 (5)
566=377+144+34+8+3 (5)
567=377+144+34+8+3+1 (6)
568=377+144+34+13 (4)
569=377+144+34+13+1 (5)
570=377+144+34+13+2 (5)
571=377+144+34+13+3 (5)
572=377+144+34+13+3+1 (6)
573=377+144+34+13+5 (5)
574=377+144+34+13+5+1 (6)
575=377+144+34+13+5+2 (6)
576=377+144+55 (3)
577=377+144+55+1 (4)
578=377+144+55+2 (4)
579=377+144+55+3 (4)
580=377+144+55+3+1 (5)
581=377+144+55+5 (4)
582=377+144+55+5+1 (5)
583=377+144+55+5+2 (5)
584=377+144+55+8 (4)
585=377+144+55+8+1 (5)
586=377+144+55+8+2 (5)
587=377+144+55+8+3 (5)
588=377+144+55+8+3+1 (6)
589=377+144+55+13 (4)
590=377+144+55+13+1 (5)
591=377+144+55+13+2 (5)
592=377+144+55+13+3 (5)
593=377+144+55+13+3+1 (6)
594=377+144+55+13+5 (5)
595=377+144+55+13+5+1 (6)
596=377+144+55+13+5+2 (6)
597=377+144+55+21 (4)
598=377+144+55+21+1 (5)
599=377+144+55+21+2 (5)
600=377+144+55+21+3 (5)
601=377+144+55+21+3+1 (6)
602=377+144+55+21+5 (5)
603=377+144+55+21+5+1 (6)
604=377+144+55+21+5+2 (6)
605=377+144+55+21+8 (5)
606=377+144+55+21+8+1 (6)
607=377+144+55+21+8+2 (6)
608=377+144+55+21+8+3 (6)
609=377+144+55+21+8+3+1 (7)
610=610 (1)
611=610+1 (2)
612=610+2 (2)
613=610+3 (2)
614=610+3+1 (3)
615=610+5 (2)
616=610+5+1 (3)
617=610+5+2 (3)
618=610+8 (2)
619=610+8+1 (3)
620=610+8+2 (3)
621=610+8+3 (3)
622=610+8+3+1 (4)
623=610+13 (2)
624=610+13+1 (3)
625=610+13+2 (3)
626=610+13+3 (3)
627=610+13+3+1 (4)
628=610+13+5 (3)
629=610+13+5+1 (4)
630=610+13+5+2 (4)
631=610+21 (2)
632=610+21+1 (3)
633=610+21+2 (3)
634=610+21+3 (3)
635=610+21+3+1 (4)
636=610+21+5 (3)
637=610+21+5+1 (4)
638=610+21+5+2 (4)
639=610+21+8 (3)
640=610+21+8+1 (4)
641=610+21+8+2 (4)
642=610+21+8+3 (4)
643=610+21+8+3+1 (5)
644=610+34 (2)
645=610+34+1 (3)
646=610+34+2 (3)
647=610+34+3 (3)
648=610+34+3+1 (4)
649=610+34+5 (3)
650=610+34+5+1 (4)
651=610+34+5+2 (4)
652=610+34+8 (3)
653=610+34+8+1 (4)
654=610+34+8+2 (4)
655=610+34+8+3 (4)
656=610+34+8+3+1 (5)
657=610+34+13 (3)
658=610+34+13+1 (4)
659=610+34+13+2 (4)
660=610+34+13+3 (4)
661=610+34+13+3+1 (5)
662=610+34+13+5 (4)
663=610+34+13+5+1 (5)
664=610+34+13+5+2 (5)
665=610+55 (2)
666=610+55+1 (3)
667=610+55+2 (3)
668=610+55+3 (3)
669=610+55+3+1 (4)
670=610+55+5 (3)
671=610+55+5+1 (4)
672=610+55+5+2 (4)
673=610+55+8 (3)
674=610+55+8+1 (4)
675=610+55+8+2 (4)
676=610+55+8+3 (4)
677=610+55+8+3+1 (5)
678=610+55+13 (3)
679=610+55+13+1 (4)
680=610+55+13+2 (4)
681=610+55+13+3 (4)
682=610+55+13+3+1 (5)
683=610+55+13+5 (4)
684=610+55+13+5+1 (5)
685=610+55+13+5+2 (5)
686=610+55+21 (3)
687=610+55+21+1 (4)
688=610+55+21+2 (4)
689=610+55+21+3 (4)
690=610+55+21+3+1 (5)
691=610+55+21+5 (4)
692=610+55+21+5+1 (5)
693=610+55+21+5+2 (5)
694=610+55+21+8 (4)
695=610+55+21+8+1 (5)
696=610+55+21+8+2 (5)
697=610+55+21+8+3 (5)
698=610+55+21+8+3+1 (6)
699=610+89 (2)
700=610+89+1 (3)
701=610+89+2 (3)
702=610+89+3 (3)
703=610+89+3+1 (4)
704=610+89+5 (3)
705=610+89+5+1 (4)
706=610+89+5+2 (4)
707=610+89+8 (3)
708=610+89+8+1 (4)
709=610+89+8+2 (4)
710=610+89+8+3 (4)
711=610+89+8+3+1 (5)
712=610+89+13 (3)
713=610+89+13+1 (4)
714=610+89+13+2 (4)
715=610+89+13+3 (4)
716=610+89+13+3+1 (5)
717=610+89+13+5 (4)
718=610+89+13+5+1 (5)
719=610+89+13+5+2 (5)
720=610+89+21 (3)
721=610+89+21+1 (4)
722=610+89+21+2 (4)
723=610+89+21+3 (4)
724=610+89+21+3+1 (5)
725=610+89+21+5 (4)
726=610+89+21+5+1 (5)
727=610+89+21+5+2 (5)
728=610+89+21+8 (4)
729=610+89+21+8+1 (5)
730=610+89+21+8+2 (5)
731=610+89+21+8+3 (5)
732=610+89+21+8+3+1 (6)
733=610+89+34 (3)
734=610+89+34+1 (4)
735=610+89+34+2 (4)
736=610+89+34+3 (4)
737=610+89+34+3+1 (5)
738=610+89+34+5 (4)
739=610+89+34+5+1 (5)
740=610+89+34+5+2 (5)
741=610+89+34+8 (4)
742=610+89+34+8+1 (5)
743=610+89+34+8+2 (5)
744=610+89+34+8+3 (5)
745=610+89+34+8+3+1 (6)
746=610+89+34+13 (4)
747=610+89+34+13+1 (5)
748=610+89+34+13+2 (5)
749=610+89+34+13+3 (5)
750=610+89+34+13+3+1 (6)
751=610+89+34+13+5 (5)
752=610+89+34+13+5+1 (6)
753=610+89+34+13+5+2 (6)
754=610+144 (2)
755=610+144+1 (3)
756=610+144+2 (3)
757=610+144+3 (3)
758=610+144+3+1 (4)
759=610+144+5 (3)
760=610+144+5+1 (4)
761=610+144+5+2 (4)
762=610+144+8 (3)
763=610+144+8+1 (4)
764=610+144+8+2 (4)
765=610+144+8+3 (4)
766=610+144+8+3+1 (5)
767=610+144+13 (3)
768=610+144+13+1 (4)
769=610+144+13+2 (4)
770=610+144+13+3 (4)
771=610+144+13+3+1 (5)
772=610+144+13+5 (4)
773=610+144+13+5+1 (5)
774=610+144+13+5+2 (5)
775=610+144+21 (3)
776=610+144+21+1 (4)
777=610+144+21+2 (4)
778=610+144+21+3 (4)
779=610+144+21+3+1 (5)
780=610+144+21+5 (4)
781=610+144+21+5+1 (5)
782=610+144+21+5+2 (5)
783=610+144+21+8 (4)
784=610+144+21+8+1 (5)
785=610+144+21+8+2 (5)
786=610+144+21+8+3 (5)
787=610+144+21+8+3+1 (6)
788=610+144+34 (3)
789=610+144+34+1 (4)
790=610+144+34+2 (4)
791=610+144+34+3 (4)
792=610+144+34+3+1 (5)
793=610+144+34+5 (4)
794=610+144+34+5+1 (5)
795=610+144+34+5+2 (5)
796=610+144+34+8 (4)
797=610+144+34+8+1 (5)
798=610+144+34+8+2 (5)
799=610+144+34+8+3 (5)
800=610+144+34+8+3+1 (6)
801=610+144+34+13 (4)
802=610+144+34+13+1 (5)
803=610+144+34+13+2 (5)
804=610+144+34+13+3 (5)
805=610+144+34+13+3+1 (6)
806=610+144+34+13+5 (5)
807=610+144+34+13+5+1 (6)
808=610+144+34+13+5+2 (6)
809=610+144+55 (3)
810=610+144+55+1 (4)
811=610+144+55+2 (4)
812=610+144+55+3 (4)
813=610+144+55+3+1 (5)
814=610+144+55+5 (4)
815=610+144+55+5+1 (5)
816=610+144+55+5+2 (5)
817=610+144+55+8 (4)
818=610+144+55+8+1 (5)
819=610+144+55+8+2 (5)
820=610+144+55+8+3 (5)
821=610+144+55+8+3+1 (6)
822=610+144+55+13 (4)
823=610+144+55+13+1 (5)
824=610+144+55+13+2 (5)
825=610+144+55+13+3 (5)
826=610+144+55+13+3+1 (6)
827=610+144+55+13+5 (5)
828=610+144+55+13+5+1 (6)
829=610+144+55+13+5+2 (6)
830=610+144+55+21 (4)
831=610+144+55+21+1 (5)
832=610+144+55+21+2 (5)
833=610+144+55+21+3 (5)
834=610+144+55+21+3+1 (6)
835=610+144+55+21+5 (5)
836=610+144+55+21+5+1 (6)
837=610+144+55+21+5+2 (6)
838=610+144+55+21+8 (5)
839=610+144+55+21+8+1 (6)
840=610+144+55+21+8+2 (6)
841=610+144+55+21+8+3 (6)
842=610+144+55+21+8+3+1 (7)
843=610+233 (2)
844=610+233+1 (3)
845=610+233+2 (3)
846=610+233+3 (3)
847=610+233+3+1 (4)
848=610+233+5 (3)
849=610+233+5+1 (4)
850=610+233+5+2 (4)
851=610+233+8 (3)
852=610+233+8+1 (4)
853=610+233+8+2 (4)
854=610+233+8+3 (4)
855=610+233+8+3+1 (5)
856=610+233+13 (3)
857=610+233+13+1 (4)
858=610+233+13+2 (4)
859=610+233+13+3 (4)
860=610+233+13+3+1 (5)
861=610+233+13+5 (4)
862=610+233+13+5+1 (5)
863=610+233+13+5+2 (5)
864=610+233+21 (3)
865=610+233+21+1 (4)
866=610+233+21+2 (4)
867=610+233+21+3 (4)
868=610+233+21+3+1 (5)
869=610+233+21+5 (4)
870=610+233+21+5+1 (5)
871=610+233+21+5+2 (5)
872=610+233+21+8 (4)
873=610+233+21+8+1 (5)
874=610+233+21+8+2 (5)
875=610+233+21+8+3 (5)
876=610+233+21+8+3+1 (6)
877=610+233+34 (3)
878=610+233+34+1 (4)
879=610+233+34+2 (4)
880=610+233+34+3 (4)
881=610+233+34+3+1 (5)
882=610+233+34+5 (4)
883=610+233+34+5+1 (5)
884=610+233+34+5+2 (5)
885=610+233+34+8 (4)
886=610+233+34+8+1 (5)
887=610+233+34+8+2 (5)
888=610+233+34+8+3 (5)
889=610+233+34+8+3+1 (6)
890=610+233+34+13 (4)
891=610+233+34+13+1 (5)
892=610+233+34+13+2 (5)
893=610+233+34+13+3 (5)
894=610+233+34+13+3+1 (6)
895=610+233+34+13+5 (5)
896=610+233+34+13+5+1 (6)
897=610+233+34+13+5+2 (6)
898=610+233+55 (3)
899=610+233+55+1 (4)
900=610+233+55+2 (4)
901=610+233+55+3 (4)
902=610+233+55+3+1 (5)
903=610+233+55+5 (4)
904=610+233+55+5+1 (5)
905=610+233+55+5+2 (5)
906=610+233+55+8 (4)
907=610+233+55+8+1 (5)
908=610+233+55+8+2 (5)
909=610+233+55+8+3 (5)
910=610+233+55+8+3+1 (6)
911=610+233+55+13 (4)
912=610+233+55+13+1 (5)
913=610+233+55+13+2 (5)
914=610+233+55+13+3 (5)
915=610+233+55+13+3+1 (6)
916=610+233+55+13+5 (5)
917=610+233+55+13+5+1 (6)
918=610+233+55+13+5+2 (6)
919=610+233+55+21 (4)
920=610+233+55+21+1 (5)
921=610+233+55+21+2 (5)
922=610+233+55+21+3 (5)
923=610+233+55+21+3+1 (6)
924=610+233+55+21+5 (5)
925=610+233+55+21+5+1 (6)
926=610+233+55+21+5+2 (6)
927=610+233+55+21+8 (5)
928=610+233+55+21+8+1 (6)
929=610+233+55+21+8+2 (6)
930=610+233+55+21+8+3 (6)
931=610+233+55+21+8+3+1 (7)
932=610+233+89 (3)
933=610+233+89+1 (4)
934=610+233+89+2 (4)
935=610+233+89+3 (4)
936=610+233+89+3+1 (5)
937=610+233+89+5 (4)
938=610+233+89+5+1 (5)
939=610+233+89+5+2 (5)
940=610+233+89+8 (4)
941=610+233+89+8+1 (5)
942=610+233+89+8+2 (5)
943=610+233+89+8+3 (5)
944=610+233+89+8+3+1 (6)
945=610+233+89+13 (4)
946=610+233+89+13+1 (5)
947=610+233+89+13+2 (5)
948=610+233+89+13+3 (5)
949=610+233+89+13+3+1 (6)
950=610+233+89+13+5 (5)
951=610+233+89+13+5+1 (6)
952=610+233+89+13+5+2 (6)
953=610+233+89+21 (4)
954=610+233+89+21+1 (5)
955=610+233+89+21+2 (5)
956=610+233+89+21+3 (5)
957=610+233+89+21+3+1 (6)
958=610+233+89+21+5 (5)
959=610+233+89+21+5+1 (6)
960=610+233+89+21+5+2 (6)
961=610+233+89+21+8 (5)
962=610+233+89+21+8+1 (6)
963=610+233+89+21+8+2 (6)
964=610+233+89+21+8+3 (6)
965=610+233+89+21+8+3+1 (7)
966=610+233+89+34 (4)
967=610+233+89+34+1 (5)
968=610+233+89+34+2 (5)
969=610+233+89+34+3 (5)
970=610+233+89+34+3+1 (6)
971=610+233+89+34+5 (5)
972=610+233+89+34+5+1 (6)
973=610+233+89+34+5+2 (6)
974=610+233+89+34+8 (5)
975=610+233+89+34+8+1 (6)
976=610+233+89+34+8+2 (6)
977=610+233+89+34+8+3 (6)
978=610+233+89+34+8+3+1 (7)
979=610+233+89+34+13 (5)
980=610+233+89+34+13+1 (6)
981=610+233+89+34+13+2 (6)
982=610+233+89+34+13+3 (6)
983=610+233+89+34+13+3+1 (7)
984=610+233+89+34+13+5 (6)
985=610+233+89+34+13+5+1 (7)
986=610+233+89+34+13+5+2 (7)
987=987 (1)

…

(the following are some large numbers, showing the position, not value, of the 2-fib numbers summing to them.

Just to clarify for myself here, by 'position' you refer to ordinal position as in the 87th Fibonacci number for example? If so, did you include zero as the 1st element?

1100087778366101931=88 (1)
1100087778366101930=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5+3 (43)
1100087778366101929=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5+2 (43)
1100087778366101928=87+85+83+81+79+77+75+73+71+69+67+65+63+61+59+57+55+53+51+49+47+45+43+41+39+37+35+33+31+29+27+25+23+21+19+17+15+13+11+9+7+5 (42)

Here’s the MUMPS code that generated it
s N=2 k F s F(1)=1 f A=2:1 q:F(A-1)>9999999999999999999  f I=A-N:1:A-1 s F(A)=$g(F(A))+$g(F(I)) ;build n-fib sequence
s A9=$o(F(""),-1) f B=1:1:987 w !,B s C=B,T=0 x "f A=A9:-1:1 i F(A)'>C w $s('T:""="",1:""+""),F(A) s C=C-F(A),T=T+1 q:'C" w " (",T,")" i C w !,B," no solution!" q  ;find and list sums
PS: maybe this thread should be broken into a couple of separate threads and put in the math forum? It’s become a bit unwatercoolerish :)

I can be led easier than I can be summoned. ;) :eek2:

CraigD · March 31, 2007

So in your expressions, n is the ordinal? That is to say, if I denote the 'fifth Fibonacci number' then n=5?

What does the n here in 'n-fib' take its value from? Also, 0 (zero) is not in the Fibonacci set so its unclear to me where you get the set {0,1,1,2}?

A bit of convention-setting seems in order.

The Fibonacci sequence can be generalized with the following formula:

[math]F_n(m)= \sum_{j=1}^n F_n(m-j)[/math]

and is called an n-fib sequence. Further, the sequence can be “seeded” with any sequence of starting values. An n-fib sequence were each term is as small as possible, I call “minimal”. The usual Fibonacci sequence, then, is a minimal 2-fib sequence.

A way to define a minimal n-fib sequence is to begin with at least n-1 zeros, followed by a 1, so,

a minimal 2-fib sequence is {0,1,1,2,3,5,8,13,21,34 …}
a minimal 3-fib seq, {0,0,1,1,2,4,7,13,24,44 …}
a minimal 4-fib seq, {0,0,0,1,1,2,4,8,15,29,56,108,208 …}

You can remove any of the beginning terms from an n-fib sequence without changing later ones, so where you chose to begin an n-fib sequence is a matter of convention. I chose to have the first term be the second 1, so [math]F_n(1)=1[/math] and [math]F_n(2)=2[/math] for all n, while [math]F_n(3)[/math] and subsequent terms depend on n. Per this convention, the minimal 2-fib sequence is {1,2,3,5,8,13 …}, [math]F_2(87)[/math]=1100087778366101931, [math]F_3(69)[/math]=1127444240280152749, [math]F_4(65)[/math]=1899425365020742591, [math]F_{99}(61)[/math]=1152921504606846976

In post #38, I actually started 1 term earlier, counting [math]F_2(1)=1[/math] and [math]F_2(2)=1[/math]. I rechecked using the [math]F_n(1)=1[/math], [math]F_n(2)=2[/math] convention, and it appears to have made no difference in finding a counterexample of the digbog conjecture.

Turtle · April 1, 2007

A bit of convention-setting seems in order.

The Fibonacci sequence can be generalized with the following formula:

[math]F_n(m)= \sum_{j=1}^n F_n(m-j)[/math]

and is called an n-fib sequence. Further, the sequence can be “seeded” with any sequence of starting values. An n-fib sequence were each term is as small as possible, I call “minimal”. The usual Fibonacci sequence, then, is a minimal 2-fib sequence.

A way to define a minimal n-fib sequence is to begin with at least n-1 zeros, followed by a 1, so,
a minimal 2-fib sequence is {0,1,1,2,3,5,8,13,21,34 …}
a minimal 3-fib seq, {0,0,1,1,2,4,7,13,24,44 …}
a minimal 4-fib seq, {0,0,0,1,1,2,4,8,15,29,56,108,208 …}
You can remove any of the beginning terms from an n-fib sequence without changing later ones, so where you chose to begin an n-fib sequence is a matter of convention. I chose to have the first term be the second 1, so [math]F_n(1)=1[/math] and [math]F_n(2)=2[/math] for all n, while [math]F_n(3)[/math] and subsequent terms depend on n. Per this convention, the minimal 2-fib sequence is {1,2,3,5,8,13 …}, [math]F_2(87)[/math]=1100087778366101931, [math]F_3(69)[/math]=1127444240280152749, [math]F_4(65)[/math]=1899425365020742591, [math]F_{99}(61)[/math]=1152921504606846976

In post #38, I actually started 1 term earlier, counting [math]F_2(1)=1[/math] and [math]F_2(2)=1[/math]. I rechecked using the [math]F_n(1)=1[/math], [math]F_n(2)=2[/math] convention, and it appears to have made no difference in finding a counterexample of the digbog conjecture.

I'm afraid I still don't understand n-fib, or why you invoke it. :doh: :shrug: You seem to use the word 'convention' differently in different places. At the top, it seems to conote a 'standard', wheras later it appears to conote 'free to choose'. :shrug: At any rate, if the general form is just an expansion of the Fibonacci Numbers as 'usually' given, i.e. {1 1 2 3 4 5 6...}, than I (we?) can easier follow accepting that form as the 'standard'.

If the n-fib explanation is as simple as it gets, then I'll keep re-reading it & see if it doesn't eventually soak in, but if you have further elucidation I'll be happy to read that as well. :)

Meanwhile, we have fallen afield of the perfect/figurate discussion and I'm particularly interested in what you think about my comment in post #22 about mathworld mistaking a statement. Feel free to split things up as you see fit, and I'll continue to fit things as I see split free. ;) :turtle:

Turtle · April 1, 2007

Just me again. :doh: :eek2: I don't want to put the onus just on Craig, so if any of you think you see a point that will help alleviate my lack of understanding here, by all means let 'er rip.

One thing further came to mind as to the 'usual' Fibonacci set, and that is succesive pairs ever more closely approach the Golden Mean in their ratio of larger to smaller. This doesn't seem to be the case with the other n-fibs Craig constructed. Is this a feature unique to the usual Fibonacci set {1 1 2 3 4 5 8 11 13...}? :) :turtle:

CraigD · April 1, 2007

I'm afraid I still don't understand n-fib, or why you invoke it.
…
if you have further elucidation I'll be happy to read that as well. :)

The idea of a generalized, “n-fib” Fibonacci sequence comes simply from the realization that “add the previous 2 terms to get the next” isn’t the only possible way to generate something like the Fibonacci sequence. “Add the previous 3 terms” (3-fib), or “add the previous 100 terms” (100-fib) will work too, and give interesting sequences with properties similar to the “classic” Fibinacci sequence (a 2-fib sequence). In particular, the digbog conjecture appears to be true for any n-fib sequence, as long as it begins with the smallest possible terms.

An n-fib sequence can start with any sequence of n numbers, but its usual to start them with the smallest possible non-negative integers. Though we usually write the Fibonacci sequence (2-fib) as {1,1,2,3,5,8,13,21,34,55 …}, it can actually be written with smaller starting non-negative integers: {0,1,1,2,3,4,8,13,21,34 …}. You could write a 2-fib sequence with even smaller starting non-negative integers – all zeros - {0,0,0,0, …}, but it’s pretty boring. ;)

Note that the “n-1 zeros followed by a one” way of starting an n-fib sequence is the only one that gives the smallest following (non-zero) terms. The 5-fib sequence

{0,0,0,0,1,1,2,4,8,16,31,61,120 …}

has smaller terms than one starting with 5 ones

{1,1,1,1,1,5,9,17,33,65,129,253,497,977, …}

One thing further came to mind as to the 'usual' Fibonacci set, and that is succesive pairs ever more closely approach the Golden Mean in their ratio of larger to smaller. This doesn't seem to be the case with the other n-fibs Craig constructed. Is this a feature unique to the usual Fibonacci set {1 1 2 3 4 5 8 11 13...}? :cup:

As Turtle notes, the ratio of consecutive terms of any 2-fib sequence approaches the golden ratio ([math]\frac{\sqrt5 +1}2[/math]), but not that of other n-fib sequences. As n increases, the ratio of consecutive terms of any n-fib sequence approaches 2.

Meanwhile, we have fallen afield of the perfect/figurate discussion and I'm particularly interested in what you think about my comment in post #22 about mathworld mistaking a statement.

... In addition, all even perfect numbers are hexagonal numbers, so it follows that even perfect numbers are always the sum of consecutive positive integers starting at 1, for example, ...

I believe all hexagonal numbers are triangular numbers, so the mathworld article, while a bit confusing, is not incorrect.

Feel free to split things up as you see fit, and I'll continue to fit things as I see split free. ;)

I think it would be a good idea to split a “perfect number” and a “Fibonacci sequence/digbog conjecture” thread from this thread, into the math forum – they’re pretty unrelated. Watercooler mods or admins, please?

Turtle · April 1, 2007

The idea of a generalized, “n-fib” Fibonacci sequence comes simply from the realization that “add the previous 2 terms to get the next” isn’t the only possible way to generate something like the Fibonacci sequence. “Add the previous 3 terms” (3-fib), or “add the previous 100 terms” (100-fib) will work too,

Bingo!!!! Now I get it!!! :turtle: :hihi:

OK, now try this for a possible angle of proof of the digbog conjecture (where is that rascal anyway???). Staying with our 2-fib standard issue, for any F(n), F(n-1) > 1/2 F(n). Does that ring up any bells or buzzers??

I believe all hexagonal numbers are triangular numbers, so the mathworld article, while a bit confusing, is not incorrect.

I suspected it was merely misleading. Perhaps a separate thread for Figurate numbers?

I think it would be a good idea to split a “perfect number” and a “Fibonacci sequence/digbog conjecture” thread from this thread, into the math forum – they’re pretty unrelated. Watercooler mods or admins, please?

Roger; we have a Perfect Number thread here

Getting late, so will sleep on it for now. :hyper: (yeah right! It'll keep me up mores like it! :hyper: )

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