A296542 - OEIS

%I #6 Dec 17 2017 00:22:33

%S 8,15,24,16,8,10,24,52,56,29,10,10,12,35,100,160,140,56,12,12,12,14,

%T 48,177,388,498,348,110,14,14,14,14,16,63,294,833,1428,1470,854,225,

%U 16,16,16,16,16,18,80,464,1632,3532,4848,4176,2080,469,18,18,18,18,18,18,20,99,702,2979,7848,13545,15534,11493,5004,991

%N Triangle read by rows T(n,k): number of undirected cycles of length k in the n-antiprism graph (n = 3...; k = 3..2n)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>

%F Polynomials satisfy the linear recurrence

%F a(n) = (2 + 2 x + 3 x^2)*a(n-1)

%F + (-1 - 4 x - 7 x^2 - 4 x^3 - x^4)*a(n-2)

%F + (-x (-2 - 5 x - 8 x^2 - 4 x^3 - 2 x^4 + 2 x^5))*a(n-3)

%F + (x^2 (-1 - 4 x - 5 x^2 - 4 x^3 + 3 x^4))*a(n-4)

%F + (x^4 (2 + 2 x + x^6))*a(n-5)

%F - (x^6 (1 + 2 x^4))*a(n-6)

%F + x^10*a(n-7)

%e Written as cycle polynomials:

%e 8 x^3 + 15 x^4 + 24 x^5 + 16 x^6

%e 8 x^3 + 10 x^4 + 24 x^5 + 52 x^6 + 56 x^7 + 29 x^8

%e 10 x^3 + 10 x^4 + 12 x^5 + 35 x^6 + 100 x^7 + 160 x^8 + 140 x^9 + 56 x^10

%e ...

%e giving the array

%e 8, 15, 24, 16;

%e 8, 10, 24, 52, 56, 29;

%e 10, 10, 12, 35, 100, 160, 140, 56;

%e ...

%t CoefficientList[LinearRecurrence[{2 + 2 x + 3 x^2, -1 - 4 x - 7 x^2 - 4 x^3 - x^4, -x (-2 - 5 x - 8 x^2 - 4 x^3 - 2 x^4 + 2 x^5), x^2 (-1 - 4 x - 5 x^2 - 4 x^3 + 3 x^4), x^4 (2 + 2 x + x^6), -x^6 (1 + 2 x^4), x^10}, {8 x^3 + 15 x^4 + 24 x^5 + 16 x^6, 8 x^3 + 10 x^4 + 24 x^5 + 52 x^6 + 56 x^7 + 29 x^8, 10 x^3 + 10 x^4 + 12 x^5 + 35 x^6 + 100 x^7 + 160 x^8 + 140 x^9 + 56 x^10, 12 x^3 + 12 x^4 + 12 x^5 + 14 x^6 + 48 x^7 + 177 x^8 + 388 x^9 + 498 x^10 + 348 x^11 + 110 x^12, 14 x^3 + 14 x^4 + 14 x^5 + 14 x^6 + 16 x^7 + 63 x^8 + 294 x^9 + 833 x^10 + 1428 x^11 + 1470 x^12 + 854 x^13 + 225 x^14, 16 x^3 + 16 x^4 + 16 x^5 + 16 x^6 + 16 x^7 + 18 x^8 + 80 x^9 + 464 x^10 + 1632 x^11 + 3532 x^12 + 4848 x^13 + 4176 x^14 + 2080 x^15 + 469 x^16, 18 x^3 + 18 x^4 + 18 x^5 + 18 x^6 + 18 x^7 + 18 x^8 + 20 x^9 + 99 x^10 + 702 x^11 + 2979 x^12 + 7848 x^13 + 13545 x^14 + 15534 x^15 + 11493 x^16 + 5004 x^17 + 991 x^18}, 10]/x^3, x] // Flatten

%K nonn,tabf

%O 1,1

%A _Eric W. Weisstein_, Dec 15 2017