An example taken from the new "sequences window" of FAMP
creigh at o2online.de
creigh at o2online.de
Mon Oct 18 00:17:05 CEST 2004
Dear Seqfans,
Below is the new "sequences window" from an updated (beta) version of FAMP.
The example given was the first found connecting the floretions
to Fibonnaci numbers without using one of the "Emmy's three" floretions;
instead, it uses a (also quite special) floretion from "Chu's group". Just
to mention two sequences, see for ex. "jesseq" for N*H or "achuseq" for
H*N, below (actually, it appears there is a whole mosaic of such connections
if one also takes, for example, bisections of the sequences into account).
Does anyone understand the posseq/negseq of N*H? Unfortunately, I
don't. It would also be nice to hear if anyone has a suggestion for new
symmetries (especially dynamic ones, as described in my last post).
If I include your symmetry, I will, of course, credit you in the program.
Thanks!
Sincerely,
Creighton
Results for the floretion: N*H
*********************************************************************************************************************************************
**********************
0.0'i - 1.5'j - 1.0'k + 0.0i' + 1.0j' - 1.0k' + 0.0'ii' - 0.5'jj' + 0.0'kk'
- 1.0'ij' - 1.0'ik' - 0.5'ji' - 0.5'jk' + 1.0'ki' + 0.0'kj' + 0.0e
*********************************************************************************************************************************************
**********************
N = 0.0 'i + 1.5 'j + 1.0 'k - 1.0 i' + 0.0 j' + 1.0 k' - 1.0 'ii' + 0.5
'jj' + 0.0 'kk' + 1.0 'ij' - 1.0 'ik' + 0.5 'ji' + 0.5 'jk' + 1.0 'ki' -
1.0
'kj' + 0.0 e
H = .5 'i + .5 'ii' + .5 'ij' + .5 'ik' +
Invariant to swap operator: no, as it does not commute with the floretion: 'ii'
+ 'jj' + 'kk' - 1
N commutes with H: no
**************************** Static Symmetries
***************************************************************************************************************
(squaring symmetries:)
vesseq: -5, -4, 15, 11, -40, -29, 105, 76, -275, -199, 720, 521, -1885, -1364,
4935, 3571
2tesseq: 0, -3, 0, 7, 0, -18, 0, 47, 0, -123, 0, 322, 0, -843, 0, 2207
2lesseq: -5, 0, 20, 0, -55, 0, 145, 0, -380, 0, 995, 0, -2605, 0, 6820,
0
2jesseq: -5, -5, 10, 15, -25, -40, 65, 105, -170, -275, 445, 720, -1165, -1885,
3050, 4935
identity: ves = jes + les + tes
(Emmy's Three, subgroup symmetries:)
em[I]seq: -1, -1, 4, 2, -11, -5, 29, 13, -76, -34, 199, 89, -521, -233, 1364,
610
em[I*]seq: -4, -3, 11, 9, -29, -24, 76, 63, -199, -165, 521, 432, -1364,
-1131, 3571, 2961
em[J]seq: -4, -4, 14, 11, -38, -29, 100, 76, -262, -199, 686, 521, -1796, -1364,
4702, 3571
em[J*]seq: -1, 0, 1, 0, -2, 0, 5, 0, -13, 0, 34, 0, -89, 0, 233, 0
em[K]seq: -1, -2, 1, 5, -2, -13, 5, 34, -13, -89, 34, 233, -89, -610, 233,
1597
em[K*]seq: -4, -2, 14, 6, -38, -16, 100, 42, -262, -110, 686, 288, -1796,
-754, 4702, 1974
2famseq: -1, -3, 4, 7, -11, -18, 29, 47, -76, -123, 199, 322, -521, -843,
1364, 2207
2fam*seq: -9, -5, 26, 15, -69, -40, 181, 105, -474, -275, 1241, 720, -3249,
-1885, 8506, 4935
identities: em[I] + em[J] + em[K] = 2*fam + ves
em[I] + em[I*] = em[J] + em[J*] = em[K] + em[K*] = fam + fam* = ves
(Chu's group, subgroup symmetries, incomplete:)
2chuseq: -9, -6, 32, 18, -87, -48, 229, 126, -600, -330, 1571, 864, -4113,
-2262, 10768, 5922
2(signed)chuseq: -9, -8, 36, 24, -99, -64, 261, 168, -684, -440, 1791,
1152, -4689, -3016, 12276, 7896
2chutesseq: -9, -9, 32, 25, -87, -66, 229, 173, -600, -453, 1571, 1186,
-4113, -3105, 10768, 8129
2achuseq: -1, 1, -2, -3, 7, 8, -19, -21, 50, 55, -131, -144, 343, 377,
-898, -987
**************************** Dynamic Symmetries
***********************************************************************************************************
2posseq: 4, 7, 48, 43, 50, 56, 342, 299, 346, 385, 2346, 2050, 2372, 2639,
16080, 14051
2negseq: -14, -15, -18, -21, -130, -114, -132, -147, -896, -783, -906,
-1008, -6142, -5367, -6210, -6909
identity: pos + neq = ves
Results for the floretion: H*N
*********************************************************************************************************************************************
**********************
0.0'i + 1.5'j + 1.0'k - 1.0i' + 0.0j' + 1.0k' - 1.0'ii' + 0.5'jj' + 0.0'kk'
+ 0.0'ij' - 1.0'ik' + 0.5'ji' + 0.5'jk' + 0.0'ki' - 1.0'kj' + 0.0e
*********************************************************************************************************************************************
**********************
N = 0.0 'i + 1.5 'j + 1.0 'k - 1.0 i' + 0.0 j' + 1.0 k' - 1.0 'ii' + 0.5
'jj' + 0.0 'kk' + 1.0 'ij' - 1.0 'ik' + 0.5 'ji' + 0.5 'jk' + 1.0 'ki' -
1.0
'kj' + 0.0 e
H = .5 'i + .5 'ii' + .5 'ij' + .5 'ik' +
Invariant to swap operator: no, as it does not commute with the floretion: 'ii'
+ 'jj' + 'kk' - 1
N commutes with H: no
**************************** Static Symmetries
***************************************************************************************************************
(squaring symmetries:)
vesseq: 1, -4, -7, 11, 20, -29, -53, 76, 139, -199, -364, 521, 953, -1364,
-2495, 3571
2tesseq: 0, -3, 0, 7, 0, -18, 0, 47, 0, -123, 0, 322, 0, -843, 0, 2207
2lesseq: -3, 0, 0, 0, 3, 0, -9, 0, 24, 0, -63, 0, 165, 0, -432, 0
2jesseq: 5, -5, -14, 15, 37, -40, -97, 105, 254, -275, -665, 720, 1741,
-1885, -4558, 4935
identity: ves = jes + les + tes
(Emmy's Three, subgroup symmetries:)
em[I]seq: -2, 0, 4, -1, -10, 3, 26, -8, -68, 21, 178, -55, -466, 144, 1220,
-377
em[I*]seq: 3, -4, -11, 12, 30, -32, -79, 84, 207, -220, -542, 576, 1419,
-1508, -3715, 3948
em[J]seq: 2, -2, -10, 5, 28, -13, -74, 34, 194, -89, -508, 233, 1330, -610,
-3482, 1597
em[J*]seq: -1, -2, 3, 6, -8, -16, 21, 42, -55, -110, 144, 288, -377, -754,
987, 1974
em[K]seq: 0, -1, -1, 2, 3, -5, -8, 13, 21, -34, -55, 89, 144, -233, -377,
610
em[K*]seq: 1, -3, -6, 9, 17, -24, -45, 63, 118, -165, -309, 432, 809, -1131,
-2118, 2961
2famseq: -1, 1, 0, -5, 1, 14, -3, -37, 8, 97, -21, -254, 55, 665, -144, -1741
2fam*seq: 3, -9, -14, 27, 39, -72, -103, 189, 270, -495, -707, 1296, 1851,
-3393, -4846, 8883
identities: em[I] + em[J] + em[K] = 2*fam + ves
em[I] + em[I*] = em[J] + em[J*] = em[K] + em[K*] = fam + fam* = ves
(Chu's group, subgroup symmetries, incomplete:)
2chuseq: 3, -6, -16, 18, 45, -48, -119, 126, 312, -330, -817, 864, 2139,
-2262, -5600, 5922
2(signed)chuseq: 7, 0, -28, 0, 77, 0, -203, 0, 532, 0, -1393, 0, 3647,
0, -9548, 0
2chutesseq: 3, -9, -16, 25, 45, -66, -119, 173, 312, -453, -817, 1186,
2139, -3105, -5600, 8129
2achuseq: -1, 1, 2, -3, -5, 8, 13, -21, -34, 55, 89, -144, -233, 377,
610, -987
**************************** Dynamic Symmetries
***********************************************************************************************************
2posseq: 10, 7, 26, 43, 110, 56, 184, 299, 760, 385, 1262, 2050, 5210, 2639,
8650, 14051
2negseq: -8, -15, -40, -21, -70, -114, -290, -147, -482, -783, -1990,
-1008, -3304, -5367, -13640, -6909
identity: pos + neq = ves
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