%I #24 Mar 17 2024 11:24:13
%S 0,320,103493760,1989020096512,18004320077137920,
%T 119822580205402103808,679908187040469153808384,
%U 3502748255987493030839058432,16926129866817207966343976976384,78226597001366370548567920133275648,350205926622184430366093984866429304832
%N Number of 6-cycles in the n-Keller graph.
%H Andrew Howroyd, <a href="/A300849/b300849.txt">Table of n, a(n) for n = 1..100</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KellerGraph.html">Keller Graph</a>
%H <a href="/index/Rec#order_73">Index entries for linear recurrences with constant coefficients</a>, order 73.
%F Terms satisfy an order-73 linear recurrence (with very large coefficients). - _Eric W. Weisstein_, Mar 20 2018.
%F a(n) = (4^(-1 + n)*(11*2^(1 + 6*n) - 4*3^n - 2^(1 + 8*n)*3^(1 + n) - 5*2^(1 + 4*n)*3^(2 + n) - 8*3^(1 + 2*n) + 19*2^(1 + 2*n)*3^(1 + 3*n) - 17*3^(1 + 4*n) - 2*3^(1 + 5*n) + 49*3^(1 + 2*n)*4^n + 5*3^(1 + 4*n)*4^n + 4^(1 + n) + 3^(2 + n)*4^(1 + n) + 3^(1 + n)*4^(2 + 3*n) + 4*7^n - 9*2^(1 + 2*n)*7^n + 3*2^(1 + 4*n)*7^n + 17*3^(1 + n)*7^n - 2^(3 + 2*n)*3^(1 + n)*7^n + 8*3^(1 + 2*n)*7^n - 35*3^(1 + 2*n)*16^n - 16^(1 + n) - 9*17^n - 103*27^n - 5*4^(1 + 2*n)*27^n - 3*49^n + 5*3^(1 + 2*n)*64^n + 183^n - 9*256^n + 1024^n))/3 - (4^(-1 + n)*n*(-3333960*61^n + 549*7^n*(-40914 + 19845*2^(1 + 2*n) - 53473*3^n) + 62769*(10*3^(3 + 4*n) - 4*27^(1 + n)*(-14 + 5*4^n) + 9^(1 + n)*(167 - 153*2^(1 + 2*n) + 15*4^(1 + 2*n)) - 2*3^n*(277 + 207*4^(1 + n) - 891*16^n + 135*64^n) + 54*(7 + 4^n + 3*4^(1 + 2*n) - 8^(1 + 2*n) + 256^n)) + 61*n*(-90*7^n*(5373 + 1400*3^n + 1296*n) + 343*(20*3^(4 + 3*n) - 10*3^(3 + 2*n)*(-23 + 9*4^n - 6*n) + 2*3^n*(1603 + 405*2^(1 + 4*n) + 1752*n + 365*n^2 - 135*4^n*(29 + 6*n)) + 27*(-51 - 15*64^n + 16*n + 51*n^2 + 6*n^3 + 5*16^n*(21 + 4*n) - 3*4^n*(25 + n*(38 + 5*n)))))))/1694763. - _Eric W. Weisstein_, Mar 20 2018
%o (PARI) \\ Requires G function from A300818
%o \\ this takes a few seconds per term
%o seq(n)={my(q2=G(n,2,[0..3])*4, q3=G(n,3,[0..15])*6, q4=G(n,4,[0..63])*8, q6=G(n,6,[0..1023])*12, diamonds=G(n,6,apply(t->bitor(t,bitand(t,12)<<6),[0..63]))*12);
%o vector(n, n, (q6[n] - (6*q4[n]*q2[n] + 3*q3[n]^2)/4^n + 6*q4[n] + 9*diamonds[n] + 7*q2[n]^3/16^n - 12*q2[n]^2/4^n - 4*q3[n] + 4*q2[n])/12)
%o } \\ _Andrew Howroyd_, Mar 16 2018
%Y Cf. A300818 (3-cycles), A300842 (4-cycles), A300848 (5-cycles).
%K nonn
%O 1,2
%A _Eric W. Weisstein_, Mar 13 2018
%E Terms a(7) and beyond from _Andrew Howroyd_, Mar 16 2018