The 1/89 lemma

[tetrahedron]

Finally found some atomic nucleus relations to the Golden Ratio, linked to the (semi)magic numbers. Wonder whether your lemma might also?

 

Jess Tauber

[tetrahedron]

Looks like others have thought of this before me, but have been wondering whether the 11 dimensional string theory might be based on Pascal Triangle horizontals which are all powers of 11, when the individual digits are read as powers of 10. Then all the weird multidimensional objects in the 11th dimension might be manifestations of some effects from Pascal- along the different diagonal types, etc.?

 

Jess Tauber

[CraigD]

I’ve been work-consumed (and will be for a while), so haven’t given as much thought to this thread’s subject of Fibonacci sequence and Pascal triangle wonders as I’d have liked. Just casually poking at them spewed out about 500 lines of notes, though, which I’ll try to condense into a single burp/post here.

 

A sister Pascal Triangle is, like the Classical version, created by drawing integers down the sides of an open-bottomed equilateral triangle (though of course any triangle will do), and then summing any two neighboring terms in the same row and putting the result centered beneath them in the next row down. I COINED the term because I can't find any literature on these either, so I don't know what the professional terminology is.

I’ve never seen a term for these, either, so herby adopt yours, Jess :thumbs_up

 

After a bit of play, it seems clear to me that the shallow diagonals of the sister Pascal triangles give the 2-fib sequences for “non classical” initial terms other than 0,1 or 1,1. For example, the shallow diagonals (which, the left-trimmed way I’ve drawn them, are just ordinary diagonals) for

2 3

2 5 3

2 7 8 3

2 9 15 11 3

2 11 24 26 14 3

2 13 35 50 40 17 3

2 15 48 85 90 57 20 3

2 17 63 133 175 147 77 23 3

2 19 80 196 308 322 224 100 26 3

...

sum to give the 2-fib sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131 ...

 

The classical 2-fib sequences (starting 1, 1) and the Lucas sequences (1,2) are just special cases of the general 2-fib sequence starting a,b. For all cases,

 

[math]\lim_{n \to \infty} \frac{G(n+1)}{G(n)} = \frac{1 +\sqrt{5}}{2} = \varphi[/math]

 

the famous golden ratio :)

 

By the way, I was looking into the way that sister Pascal triangles should stack one atop the other in a third dimension.

…

I wonder whether some of the things you've found here might map to a larger system like this (and it might be higher-D than 3).

I haven’t been able to imagine any interesting way to stack sister triangles, but there’s a well-known generalization of Pascal’s triangle into 3-dimensions, Pascal's pyramid, which looks like this:

1  1    1      1        1            1                1   
  1 1  2 2    3 3      4 4          5  5             6  6     
       1 2 1  3 6 3    6 12 6       10 20 10         15 30 15    
              1 3 3 1  4 12 12 4    10 30 30 10      20 60 60 20    
                       1 4  6  4 1  5  20 30 20 5    15 60 90 60 15 
                                    1  5  10 10 5 1  6  30 60 60 30 6 
                                                     1  6  15 20 15 6 1

By noting that the 2-d P triangle is just an arrangement of the binomial sequence, and the pyramid of the trinomial sequence, we can draw generalized “Pascal’s hyper-pyramids” of any number of dimensions by calculating and arranging the coefficients of [imath](a+b+\dots)^n[/imath] Here’s the tip of the 4-d one:

1  1
  1 1
	
1  2    1
  2 2  2 2
       1 2 1

1  3    3      1
  3 3  6 6    3 3
       3 6 3  3 6 3
              1 3 3 1

1  4    6        4           1  
  4 4  12 12    12 12       4 4  
       6  12 6  12 24 12    6 12 6  
                4  12 12 4  4 12 12 4  
                            1 4  6  4 1

1  5    10        10           5              1              
  5 5  20 20     30 30        20 20          5  5           
       10 20 10  30 60 30     30 60 30       10 20 10       
                 10 30 30 10  20 60 60 20    10 30 30 10    
                              5  20 30 20 5  5  20 30 20 5  
                                             1  5  10 10 5 1

1  6    15        20            15                6                    1 
  6 6  30 30     60 60         60 60             30 30                6  6 
       15 30 15  60 120 60     90 180 90         60 120 60            15 30 15 
                 20 60  60 20  60 180 180 60     60 180 180 60        20 60 60 20 
                               15 60  90  60 15  30 120 180 120 30    15 60 90 60 15 
                                                 6  30  60  60  30 6  6  30 60 60 30 6 
                                                                      1  6  15 20 15 6 1

 

http://www.mi.sanu.ac.rs/vismath/spinadel/index.html

In just the first few pages of this wonderful paper, Vera de Spinadel's THE FAMILY OF METALLIC MEANS, I find a different kind of generalization, of the 2-fib sequence:

 

[math]\lim_{n \to \infty} \frac{G(n+1)}{G(n)} = \frac{p +\sqrt{p^2+4q}}{2}[/math]

 

Where [math]G(n)= qG(n-2) +pG(n-1)[/math]

 

What fascinates me most about this is that it opens the doors on special context values of the golden ratio (phi) that are rational. For example,

 

[math]G(n)= 2G(n-2) +G(n-1)[/math]

 

gives the sequence: 0 1 1 3 5 11 21 43 85 171 341 683 1365 2731 5461 10923 21845 43691 87381 …

 

for which [math]\lim_{n \to \infty} \frac{G(n+1)}{G(n)} = 2[/math]

 

There’s something comforting about rational magic numbers. :)

 

Some people have trouble thinking in terms of abstract equations. I'm one of them. But charts and graphs I can understand. And sequences. More intuitive I guess. So its also surprising that I don't like to create depictions.

Though long ago (ca. 1980) I studied enough math to get a BS degree in it, before and since then, I’ve been a writer of computer programs. So, while I wouldn’t say I really understand equations, or charts and graphs, I appreciate that the one can be used to write programs to generate the others.

 

I’ve long harbored the sneaking suspicion that computer have an effect on the math-inclined similar to that that would occur if you gave a helicopter, a machine gun, the training to use them, and an unlimited supply of fuel and ammo, to a child from a society who’s elders were patiently teaching him to hunt with only a spear. He’d bag an amazing amount of game, but, like my mathematical proof skills, his tracking and stalking skills would likely atrophy.

[tetrahedron]

Just found this last night after posting- surprisingly one of the authors is a chemist involved in the new 'mathematical chemistry' movement, which I'd only first heard of two nights ago. And my search had nothing to do with chemistry per se. To have the same person pop up that way is, well, spooky!

 

www.emis.de/journals/DM/v16-2/art3.pdf

 

They are calling these Triangles Pascal-like. Which makes sense given I call Lucas, etc. Fibonacci-like.

 

As for your post, first thought- try stacking the Triangles close-packed, as if made from spheres. Then more interesting straight-line intersphere connectivity- as in my tetrahedral models.

 

Tried (thinking second time's a charm) posting some of my stuff again over at physicsforums, only to have their head 'mathematician' (the mod in charge of their number theory section) delete my thread and attach a warning, telling me that my stuff was 'bunk'. Not just wrong, but no discussion of why- this after asking for help in identifying the various patternings. I specifically tried to find the one place where people weren't just asking for homework help too, who were doing their own personal research (contra their 'rules') off the mainstream. Was it something I ate? Or do they really believe the universe is closed and knowledge 'done'?

 

Jess Tauber

[tetrahedron]

Been a while here on this topic. Was just looking at the pages again when I noticed that within CraigD's sequence from above there is a repeating pattern of final digits, even at times double digit terms- what do you folks make of it? Are there other hidden structures?:

 

2 1/1

3 1/5

4 1/11

5 1/19

6 1/29

7 1/41

8 1/55

9 1/71

10 1/89

11 1/109

12 1/131

13 1/155

14 1/181

15 1/209

16 1/239

17 1/271

18 1/305

19 1/341

20 1/379

21 1/419

22 1/461

23 1/505

24 1/551

25 1/599

26 1/649

27 1/701

28 1/755

29 1/811

30 1/869

31 1/929

32 1/991

33 1/1055

34 1/1121

35 1/1189

36 1/1259

37 1/1331

38 1/1405

39 1/1481

40 1/1559

41 1/1639

42 1/1721

43 1/1805

44 1/1891

45 1/1979

46 1/2069

47 1/2161

48 1/2255

49 1/2351

50 1/2449

51 1/2549

52 1/2651

53 1/2755

54 1/2861

55 1/2969

56 1/3079

57 1/3191

58 1/3305

59 1/3421

60 1/3539

61 1/3659

62 1/3781

63 1/3905

64 1/4031

65 1/4159

66 1/4289

67 1/4421

68 1/4555

69 1/4691

70 1/4829

71 1/4969

72 1/5111

73 1/5255

74 1/5401

75 1/5549

76 1/5699

77 1/5851

78 1/6005

79 1/6161

80 1/6319

81 1/6479

82 1/6641

83 1/6805

84 1/6971

85 1/7139

86 1/7309

87 1/7481

88 1/7655

89 1/7831

90 1/8009

91 1/8189

92 1/8371

93 1/8555

94 1/8741

95 1/8929

96 1/9119

97 1/9311

98 1/9505

99 1/9701

100 1/9899

 

Jess Tauber

[tetrahedron]

So its been a while- hope all of you are well and happy.

 

Today, after leaving the problem for a long time, came back to it and was able to discover how to derive all the simpler generalized Fib sequences parallel to the way 1/89 patterns Fib numbers as terminating in successive negative powers of 10.

 

And the solution is trivially simple, which I worked out on paper with the aid of handheld calculator.

 

Just add 9/89 successively starting with 1/89. 10/89 reproduces the same Fib sequence, moved up a notch. 19/89 gives the Lucas sequence, 28/89 the next one (3,1,4...), 37/89 (4,1,5...), 46/89 (5,1,6...). Don't seem to be any exceptions for these simplest generalized Fib sequences. But I have to be able to generalize the rule to any.

 

Best regards to all,

Jess Tauber

[tetrahedron]

Update- after checking closely I've found that any general Fib sequence relates to some multiple of 1/89. If you take 'seeds' of two digits each from 0 to 9, the following holds: If the second term is held constant you always get a difference of 9/89 (so 1,1;2,1;3,1; or 1,2;2,2;3,2, etc.). If on the other hand you hold the first term constant the difference between seeds is 1/89, for adjacent seeds (so 1,1;2,1;3,1 etc.). At least between these limits there are no exceptions.

 

Now, can any of you tell me how this will work in other bases than 10???

 

Jess Tauber

[Turtle]

Been a while here on this topic. Was just looking at the pages again when I noticed that within CraigD's sequence from above there is a repeating pattern of final digits, even at times double digit terms- what do you folks make of it? Are there other hidden structures?:

 

2 1/1

3 1/5

4 1/11

5 1/19

6 1/29

7 1/41

8 1/55

9 1/71

10 1/89

11 1/109

...

 

 

 

12 1/131

13 1/155

14 1/181

15 1/209

16 1/239

17 1/271

18 1/305

19 1/341

20 1/379

21 1/419

22 1/461

23 1/505

24 1/551

25 1/599

26 1/649

27 1/701

28 1/755

29 1/811

30 1/869

31 1/929

32 1/991

33 1/1055

34 1/1121

35 1/1189

36 1/1259

37 1/1331

38 1/1405

39 1/1481

40 1/1559

41 1/1639

42 1/1721

43 1/1805

44 1/1891

45 1/1979

46 1/2069

47 1/2161

48 1/2255

49 1/2351

50 1/2449

51 1/2549

52 1/2651

53 1/2755

54 1/2861

55 1/2969

56 1/3079

57 1/3191

58 1/3305

59 1/3421

60 1/3539

61 1/3659

62 1/3781

63 1/3905

64 1/4031

65 1/4159

66 1/4289

67 1/4421

68 1/4555

69 1/4691

70 1/4829

71 1/4969

72 1/5111

73 1/5255

74 1/5401

75 1/5549

76 1/5699

77 1/5851

78 1/6005

79 1/6161

80 1/6319

81 1/6479

82 1/6641

83 1/6805

84 1/6971

85 1/7139

86 1/7309

87 1/7481

88 1/7655

89 1/7831

90 1/8009

91 1/8189

92 1/8371

93 1/8555

94 1/8741

95 1/8929

96 1/9119

97 1/9311

98 1/9505

99 1/9701

100 1/9899

 

 

Jess Tauber

 

 

I see the end-digit repeating pattern {1 5 1 9 9}. The end digit of a number x written base-10 is congruent to the digital root of x in base 11. I went through my Katabatak graphs base 11 and found a match -of sorts- for the pattern { 1 5 1 9 9}

 

In the Katabatak Powers Base 11 graph, powers of the form x ^4y-2. The repeating pattern is {1 4 9 6 5 6 9 4 1 0} Extending it...{1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0 ...} Start with 1, skip three and you get 5. skip three and you get 1. skip three and you get 9. skip three and you get 9. Rinse & repeat ad infinitum.

 

Well, that's all I got Pascal.

[Turtle]

I see the end-digit repeating pattern {1 5 1 9 9}. The end digit of a number x written base-10 is congruent to the digital root of x in base 11. I went through my Katabatak graphs base 11 and found a match -of sorts- for the pattern { 1 5 1 9 9}

 

In the Katabatak Powers Base 11 graph, powers of the form x ^4y-2. The repeating pattern is {1 4 9 6 5 6 9 4 1 0} Extending it...{1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0 ...} Start with 1, skip three and you get 5. skip three and you get 1. skip three and you get 9. skip three and you get 9. Rinse & repeat ad infinitum.

 

Well, that's all I got Pascal.

 

 

Here is an expansion of values generated by my system.

 

 

x ^(4y-2)

 

y=

1 x^2

2 x^6

3 x^ 10

4 x^ 14

...

 

x^2

n=

1 1

5 25

9 81

13 169

17 289

...

x^6

1 1

5 15625

9 531441

13 4826809

17 24137569

...

x^10

1 1

5 9765625

9 3486784401

13 137858491849

17 2015993900449

...

 

 

 

[tetrahedron]

Thanks. Not that I understand half of what you wrote :) . Way over MY head.

 

Yesterday I started looking at series similar to generalized Fibonacci, except for other Metallic Means (Fib of course relating to the Golden).

 

The Metallic Means can be calculated several ways, but I find most convenient the equation 1/2(N+(sqrt(N^2+4)). This equation yields 1.000... when N=0, the Golden Mean for N=1, Silver Mean for N=2, and so on through other 'metals'.

 

We already know that in base 10 the fundamental term for Phi-based sequences is 1/89, with differences of 1/89 when the first seed term is held constant but we vary the second one at a time (1,2,3...), and 9/89 when we vary the first but hold the second constant similarly.

 

I found that for the Silver Mean (N=2) the fundamental base 10 term is 1/79, and varying the first seed term holding the second constant gives 8/79.

For the next one after this (based on N=3) fundamental is 1/69, variable 7/69, and for N=4 fundamental 1/59, variable 6/59.

 

That is, say for the Silver Mean, to generate decimalized strings containing, over negative powers of ten, the entire series of generalized Pell numbers (analogous to generalized Fib numbers), we keep adding to the fundamental fraction increments of 8/79.

 

So all of this, at least in base 10, is a massive interconnecting system. From this perspective I want to remind the folks that I also found several years ago that the parent sequences (nondecimalized) behind these relate to the Pascal Triangle system.

 

Powers of the Metallic Means can be generated by equations each of whose terms have numerical coefficients and power values that come out of the diagonal and diagonal dimensional values of a (2,1)-sided generalized Pascal Triangle (what I used to call a 'sister' Pascal). The shallow diagonals of this Triangle, leading up to the 2's side, sample numerical values summing to Lucas numbers, and the values themselves are the numerical values of the terms giving individual powers of Metallic Means.

 

I haven't yet found it possible to create multidimensional analogues of the Pascal system whose summed shallow dimensional values give the basic sequences (like Fib or Pell) associated with each Metallic Mean. Doesn't mean impossible to do- after all I just worked out the whole 1/89 thing for these, but I'm not often as clever as I think I am.

 

There are still lots of questions- all the Metallic Means associated decimalized strings relate, in base 10, to fractions whose denominators all seem to end in 9. Does this then shift in some regular way for similar sequences in other bases? Or are such shifts part of a larger hierarchically higher similar system for Pisot-Vijayaraghavan numbers? Just how far does all this go?

 

Jess Tauber

Edited August 10, 2013 by pascal

[tetrahedron]

Please note change in username!  Anyway, I'm back on. I recently discovered the following relationship: in the harmonic oscillator-only model of nuclei, for spheres, the energy levels are all 1.5, 2.5. 3.5, 4.5, 5.5, 6.5, 7.5.... in terms of h-bar omega bar energy units. The shells have occupancies which are themselves doubled triangular numbers because of the way the orbitals assort into period analogues- only positive (s,d,g,i...) or negative parity (p,d,h,j..) quantum numbers ml.

 

1s=2

1p=6

1d2s=12

1f2p=20

1g2d3s=30

1h2f3p=42

1i2g3d4s=56

1j2h3f4p=72

 

If we multiply the energy levels with their associated shell occupancies, the resulting products are:

 

3,15,42,90,165,273,420,612,855....

 

Differences between the products are then:

 

12,27,48,75,198,147,192,243.... which, when divided by 3, become 4,9,16,25,36,49,64,81

 

If we divide the original set (3,15,42,,,,) by 3, we end up with the set:  1,3,5,14,30,55,91,140.204....

 

These last numbers then turn out to be the terms of the diagonal from the (2,1) 'sister' Pascal Triangle (which I now know is usually called a 'generalized' Pascal Triangle, which is equivalent to the tetrahedral numbers from the classical Pascal system).

 

It also turn out that these numbers are themselves SUMS of tetrahedral numbers: 5=1+4, 14=4+10, 30=10+20, 55=20+35, 91=35+56, 140=56+84, 204=84+120.. and so on.

 

Jess Tauber

Edited September 10, 2017 by yahganlang1958