A289161 Number of 3-cycles in the n X n black bishop graph.

0, 0, 2, 6, 24, 50, 114, 196, 352, 540, 850, 1210, 1752, 2366, 3234, 4200, 5504, 6936, 8802, 10830, 13400, 16170, 19602, 23276, 27744, 32500, 38194, 44226, 51352, 58870, 67650, 76880, 87552, 98736, 111554, 124950, 140184, 156066, 174002, 192660

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Black Bishop Graph

Eric Weisstein's World of Mathematics, Graph Cycle

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Formula

a(n) = (n-1)^2*(2*n^2-4*n+3-3*(-1)^n)/24.

a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8).

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

Mathematica

Table[(n - 1)^2 (2 n^2 - 4 n + 3 - 3 (-1)^n)/24, {n, 20}]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 2, 6, 24, 50, 114, 196}, 20]
CoefficientList[Series[-((2 x^2 (1 + x + 4 x^2 + x^3 + x^4))/((-1 + x)^5 (1 + x)^3)), {x, 0, 20}], x]

Prog

(PARI) concat(vector(2), Vec(2*x^3*(1 + x + 4*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3) + O(x^50))) \\ Colin Barker, Jun 27 2017

Crossrefs

Cf. A289162 (4-cycles), A289163 (5-cycles), A289160 (6-cycles).

Sequence in context: A321613 A104114 A175624 * A118038 A240428 A276838

Adjacent sequences: A289158 A289159 A289160 * A289162 A289163 A289164

Keyword

nonn,easy

Author

Eric W. Weisstein, Jun 26 2017

Status

approved