A296542 Triangle read by rows T(n,k): number of undirected cycles of length k in the n-antiprism graph (n = 3...; k = 3..2n)

8, 15, 24, 16, 8, 10, 24, 52, 56, 29, 10, 10, 12, 35, 100, 160, 140, 56, 12, 12, 12, 14, 48, 177, 388, 498, 348, 110, 14, 14, 14, 14, 16, 63, 294, 833, 1428, 1470, 854, 225, 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

Offset

1,1

Links

Table of n, a(n) for n=1..70.

Eric Weisstein's World of Mathematics, Antiprism Graph

Eric Weisstein's World of Mathematics, Cycle Polynomial

Formula

Polynomials satisfy the linear recurrence

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

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

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

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

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

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

+ x^10*a(n-7)

Example

Written as cycle polynomials:
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
...
giving the array
8, 15, 24, 16;
8, 10, 24, 52, 56, 29;
10, 10, 12, 35, 100, 160, 140, 56;
...

Mathematica

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

Crossrefs

Sequence in context: A126622 A109332 A359000 * A296546 A167986 A015727

Adjacent sequences: A296539 A296540 A296541 * A296543 A296544 A296545

Keyword

nonn,tabf

Author

Eric W. Weisstein, Dec 15 2017

Status

approved