OFFSET
0,5
LINKS
Jean-François Alcover, Table of n, a(n) for n = 0..999
A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces, J. Algebraic Combinatorics, 9 (1999), 129-140; arXiv:math/0208002 [math.CO], 2002.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Information Theory, 44 (1998), 1369-1387.
J. H. Conway, R. H. Hardin, and N. J. A. Sloane, Packing Lines, Planes, etc.: Packings in Grassmannian Space, Experimental Math. 5 (1996), 139-159; arXiv:math/0208004 [math.CO], 2002.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1).
FORMULA
Molien series = (t^148 - t^142 + t^140 + t^136 - t^134 + t^132 + 3*t^128 + 2*t^124 + 4*t^120 + 5*t^116 + 7*t^112 + t^110 + 7*t^108 + t^106 + 10*t^104 + 2*t^102 + 11*t^100 + 3*t^98 + 12*t^96 + 4*t^94 + 14*t^92 + 5*t^90 + 16*t^88 + 5*t^86 + 15*t^84 + 4*t^82 + 20*t^80 + 7*t^78 + 18*t^76 + 4*t^74 + 18*t^72 + 7*t^70 + 20*t^68 + 4*t^66 + 15*t^64 + 5*t^62 + 16*t^60 + 5*t^58 + 14*t^56 + 4*t^54 + 12*t^52 + 3*t^50 + 11*t^48 + 2*t^46 + 10*t^44 + t^42 + 7*t^40 + t^38 + 7*t^36 + 5*t^32 + 4*t^28 + 2*t^24 + 3*t^20 + t^16 - t^14 + t^12 + t^8 - t^6 + 1) /
(t^156 - t^154 - t^150 + t^148 - t^142 + t^138 + t^136 - 3*t^132 + 2*t^130 + 2*t^126 - 2*t^124 + 2*t^118 - t^116 - t^114 - t^112 + t^110 + t^108 - 2*t^106 + t^102 + t^100 - t^98 - 2*t^96 + 3*t^92 - 2*t^88 - 2*t^86 + 3*t^84 + 2*t^82 - 4*t^78 + 2*t^74 + 3*t^72 - 2*t^70 - 2*t^68 + 3*t^64 - 2*t^60 - t^58 + t^56 + t^54 - 2*t^50 + t^48 + t^46 - t^44 - t^42 - t^40 + 2*t^38 - 2*t^32 + 2*t^30 + 2*t^26 - 3*t^24 + t^20 + t^18 - t^14 + t^8 - t^6 - t^2 + 1).
MATHEMATICA
ker = {1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1};
init = {1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 18, 22, 29, 35, 48, 57, 74, 91, 116, 140, 177, 211, 265, 319, 389, 462, 566, 667, 804, 949, 1131, 1324, 1573, 1827, 2153, 2502, 2917, 3364, 3916, 4491, 5187, 5937, 6813, 7760, 8879, 10058, 11448, 12950, 14658, 16500, 18632, 20894, 23487, 26279, 29417, 32801, 36630, 40695, 45285, 50223, 55690, 61559, 68119, 75092, 82841, 91141, 100256, 110026, 120800, 132226, 144804, 158251, 172881, 188489, 205560, 223657};
LinearRecurrence[ker, init, 1000] (* Jean-François Alcover, Jan 05 2020 *)
PROG
(Magma)
// Commands to generate the group.
F<s> := QuadraticField(2); M := GeneralLinearGroup(8, F); t := 1/(2*s);
B := M! [ -t, -t, -t, -t, -t, -t, -t, -t,
-t, t, -t, -t, t, -t, t, t,
-t, -t, -t, t, -t, t, t, t,
-t, -t, t, -t, t, t, t, -t,
-t, t, -t, t, t, t, -t, -t,
-t, -t, t, t, t, -t, -t, t,
-t, t, t, t, -t, -t, t, -t,
-t, t, t, -t, -t, t, -t, t ];
S := M! [ -1, 0, 0, 0, 0, 0, 0, 0,
0, -1, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1 ];
C := M! [ 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 0, 0, 0, 0, 0, 0 ];
G := sub< M | B, S, C >; Order(G);
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, G. Nebe (nebe(AT)math.rwth-aachen.de)
EXTENSIONS
Rechecked Mar 30 2004. There were errors in the formula line, although not in the sequence itself.
STATUS
approved