A289181 Number of 6-cycles in the n X n knight graph.

0, 0, 0, 20, 164, 616, 1348, 2352, 3628, 5176, 6996, 9088, 11452, 14088, 16996, 20176, 23628, 27352, 31348, 35616, 40156, 44968, 50052, 55408, 61036, 66936, 73108, 79552, 86268, 93256, 100516, 108048, 115852, 123928, 132276, 140896, 149788, 158952, 168388, 178096

Offset

1,4

Links

Table of n, a(n) for n=1..40.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Knight Graph

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Formula

For n > 5, a(n) = 4 (34 n^2 - 259 n + 484).

G.f.: (4*x^4*(-5 - 26*x - 46*x^2 + 7*x^3 + 2*x^4))/(-1 + x)^3.

Mathematica

Table[Length[FindCycle[KnightTourGraph[n, n], {6}, All]], {n, 20}]
Table[Piecewise[{{0, n < 4}, {20, n == 4}, {164, n == 5}}, 4 (34 n^2 - 259 n + 484)], {n, 20}]
Join[{0, 0, 0, 20, 164}, LinearRecurrence[{3, -3, 1}, {1036, 408, 52}, {6, 20}]]
CoefficientList[Series[(4 x^3 (-5 - 26 x - 46 x^2 + 7 x^3 + 2 x^4))/(-1 + x)^3, {x, 0, 20}], x]

Prog

(PARI) a(n)=4*if(n>5, 34*n^2 - 259*n + 484, max(36*n-139, 0)) \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Cf. A163433 (4-cycles).

Sequence in context: A067534 A041768 A221870 * A056114 A281204 A186259

Adjacent sequences: A289178 A289179 A289180 * A289182 A289183 A289184

Keyword

nonn,easy

Author

Eric W. Weisstein, Jun 27 2017

Status

approved