"Convergents to golden ratio" sequnces, "tribonacci" sequences, and the "helious sequnce".

creigh at o2online.de creigh at o2online.de
Fri Dec 24 17:16:50 CET 2004

```Dear Seqfans,

It seems to me the sequences given, below, dealing with "convergents to golden
ratio" and (another set of sequences concerning) "tribonacci numbers" should
be of interest. I have also included a bit about the "helious sequence"
further down the page. I will submit a few of these over the next  few days.
(and many greetings!)

Results for the floretion:  I*H

****************************************************************************************************

+ .5'k + .5k' - .5'jj' + 'ij' + .5'ik' + 'ji' + .5'jk' + .5'ki' + .5'kj' +
.5e

****************************************************************************************************

I =  + 'i + i' + 'ij' + 'ji'
H =  + .5'j + .5j' + .5'ji' + .5'kj' + .5e

Invariant to swap operator: yes, as it commutes with the floretion 'ii' +
'jj' + 'kk' - 1

Note: Sequences, below, refer to I*H

1vesseq: 5, 9, 38, 112, 376, 1200, 3904, 12608, 40832, 132096, 427520, 1383424, 4476928,
14487552, 46882816, 15171584, 490962944, 1588789248, 5141430272, 16638017536,
5384175616
//unlisted

2tesseq: 1, 6, 16, 56, 176, 576, 1856, 6016, 19456, 62976, 203776,
http://www.research.att.com/projects/OEIS?Anum=A084057
Inverse binomial transform of A001077. Binomial transform of
expansion of cosh(sqrt(5)x) (1,0,5,0,25,...).

2lesseq: 7, 14, 56, 168, 560, 1792, 5824, 18816, 60928, 197120,
Convergents to golden ratio (1+sqrt(5))/2.
G.f.: 1/(1-2*x-4*x^2).
http://www.research.att.com/projects/OEIS?Anum=A063727

1jesseq: 1, -1, 2, 0, 8, 16, 64, 192, 640, 2048, 6656, 21504, 69632,
http://www.research.att.com/projects/OEIS?Anum=A087205
Inverse binomial transform of A087204

1em[I]seq: 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136,
Convergents to golden ratio (1+sqrt(5))/2

1em[I*]seq: 4, 7, 30, 88, 296, 944, 3072, 9920, 32128, 103936, 336384, 1088512,
3522560, 11399168, 36888576, 119373824, 386301952, 12500992, 4045406208,
13091209216,
// unlisted

1em[J]seq: 1, 3, 10, 32, 104, 336, 1088, 3520, 11392, 36864,
Ratio of successive terms approaches sqrt(5) + 1.
http://www.research.att.com/projects/OEIS?Anum=A063782

1em[J*]seq: 4, 6, 28, 80, 272, 864, 2816, 9088, 29440, 95232, 308224, 997376,
3227648, 104448, 33800192, 109379584, 353959936, 1145438208, 370671616,
11995185152,
//unlisted

1em[K]seq: 3, 4, 20, 56, 192, 608, 1984, 6400, 20736, 67072, 217088, 702464,
2273280, 7356416, 23805952, 77037568, 249298944, 80674816, 2610692096, 8448376832,
//unlisted

1em[K*]seq: 2, 5, 18, 56, 184, 592, 1920, 6208, 20096, 65024, 210432, 680960,
2203648, 7131136, 23076864, 74678272, 241664, 782041088, 2530738176, 8189640704,
26502234112,

1famseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
1fam*seq: 5, 9, 38, 112, 376, 1200, 3904, 12608, 40832, 132096, 427520,
1383424, 4476928, 14487552, 46882816, 15171584, 490962944, 1588789248, 5141430272,
16638017536,

identities:
em[I] + em[J] + em[K] = 2fam + ves  ;
em[I] + em[I*] = em[J] + em[J*] = em[K] + em[K*] = fam + fam* = ves  ;
jes + les + tes = ves

Note: Sequences, below, refer to H*I

1vesseq: -1, 1, -2, 0, -8, -16, -64, -192, -640, -2048, -6656, -21504, -
see jesseq(I*H), above

2tesseq: 1, 6, 16, 56, 176, 576, 1856, 6016, 19456, 62976, 203776, 659456, 2134016,
6905856, 22347776, 72318976, 234029056, 757334016, 2450784256, 7930904576,
25664946176, 83053510656

2lesseq: 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912,
see em[I]seq(I*H), above;

1jesseq: -2, -3, -14, -40, -136, -432, -1408, -4544, -14720, -47616, -154112,
-498688, -1613824, -5222400, -16900096, -54689792, -176979968, -572719104,
-185335808, -5997592576, -19408617472, -62807605248
//unlisted

2jesleftseq: -3, -5, -22, -64, -216, -688, -2240, -7232, -23424, -75776, -245248,
-793600, -2568192, -8310784, -26894336, -87031808, -28164096, -911409152,
-2949382144,  // unlisted

2jesrightseq: -1, -1, -6, -16, -56, -176, -576, -1856, -6016, -19456, -
// see tes (this sequence has an additional term)

1em[I]seq: -1, 2, 0, 8, 16, 64, 192, 640, 2048, 6656, 21504, 69632,
1em[I*]seq: 0, -1, -2, -8, -24, -80, -256, -832, -2688, -8704, -28160,
-91136, -294912, -954368,

Concerning tribonacci numbers:

Results for the floretion:  E*N

****************************************************************************************************

- .5'i - .5'j - .5'ii' + .5'jj' - .5'kk' - .5'ij' - .5'ik' + .5'ki'

****************************************************************************************************

1vesseq: -2, -1, -2, -5, -8, -15, -28, -51, -94, -173, -318, -585, -1076, -1979,
-3640, -6695, -12314,

-22649, -41658, -76621, -140928, -259207, -476756, -876891, -1612854, -2966501,
-5456246,
// unlisted

2tesseq: 0, 2, 3, 6, 10, 20, 35, 66, 120, 222, 407, 750, 1378, 2536, 4663,
8578, 15776, 29018, 53371, 98166, 180554, 332092, 610811, 1123458, 2066360,
3800630, 6990447, 12857438,
// unlisted

2lesseq: -2, -2, -3, -8, -12, -24, -43, -80, -146, -270, -495, -912, -1676,
-3084, -5671, -10432, -19186, -35290, -64907, -119384, -219580, -403872,
-742835, -1366288, -2512994, -4622118,
// unlisted

1jesseq: -1, -1, -2, -4, -7, -13, -24, -44, -81, -149, -274, -504, -927, -1705,
-3136, -5768, -10609, -19513, -35890, -66012, -121415, -223317, -410744,
-755476, -1389537, -2555757, -4700770,
http://www.research.att.com/projects/OEIS?Anum=A000073
tribonacci numbers

2jesleftseq: -2, 0, -3, -4, -8, -14, -27, -48, -90, -164, -303, -556, -1024, -1882,
-3463, -6368, -11714, -21544, -39627, -72884, -134056, -246566, -453507,
-834128, -1534202, -2821836,
// unlisted

2jesrightseq: 0, -2, -1, -4, -6, -12, -21, -40, -72, -134, -245, -452, -830,
-1528, -2809, -5168, -9504, -17482, -32153, -59140, -108774, -200068, -367981,
-676824, -1244872, -2289678,

identities:
jes + les + tes = ves
jesleft + jesright = jes

Note: the sequences "jesleft", "jesright" and "jes" are referred to as the
"jes series", in the (top) comment to
http://www.research.att.com/projects/OEIS?Anum=A001109
as well as in
http://www.research.att.com/projects/OEIS?Anum=A005251

Finally,
Regarding the "helious" sequence
http://www.research.att.com/projects/OEIS?Anum=A097357

Yesterday, I put a few observations as well as the first 3000 terms of the
sequence into the file "heliouscheck.txt" at
http://www.crowdog.de/15301.html

Notice the strings "3, 6, 3, 9, 9", "5, 10, 5, 15, 15", and "11, 22, 11" appear
to repeat an infinite number of times in the sequence. This led me to give
some thought to the term preceding any such string. I noticed the following:

Twice the term immediately preceding "3,6,3,9,9" subtracted by the term immediately

preceding the next occurance of  "3,6,3,9,9" in the sequence gives:

2*11 - 22 = 0
2*22 - 43 = 1
2*43 - 86 = 0
2*86 - 171 =  1
2*171 - 342 = 0
2*342 - 683 = 1
2*683 - 1366 = 0
2*1366 - 2731 = 1

Twice the term immediately preceding "5, 10, 5, 15, 15" subtracted by the
term immediately

preceding the next occurance of  "5, 10, 5, 15, 15" in the sequence gives:

31*2 - 64 = - 2
64*2 - 127 = 1
127*2 - 256 = -2
2*256 - 511 = 1
511*2 - 1024 = -2

// same as above with "11, 22, 11"

2*28 - 53 = 3
2*53 - 108 = -2
2*108 - 213 = 3
2*213 - 428 = -2
428*2 - 853 = 3

Would anyone be interested in making an animation out of rings in the picture
given at the link? If, in addition, the user could set the number and initial
placement of these rings (for ex., using a mouse) then that user will have
effictively created his/her own sequence as it depends on these values.

Sincerely,
Creighton

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