A289163 Number of 5-cycles in the n X n black bishop graph.

0, 0, 0, 10, 84, 322, 1172, 2780, 7016, 13532, 27720, 47318, 84796, 133294, 217756, 322392, 492240, 696216, 1009680, 1377474, 1918500, 2541946, 3426852, 4431988, 5816888, 7371572, 9460568, 11782862, 14837004, 18204326, 22551340, 27310384, 33355168, 39932592, 48168480

Offset

1,4

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 (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Formula

a(n) = (-165 - 282*n + 972*n^2 - 730*n^3 + 280*n^4 - 68*n^5 + 8*n^6 - 5*(-1)^n*(-33 - 26*n + 96*n^2 - 46*n^3 + 6*n^4))/240.

a(n) = 2*a(n-1)+4*a(n-2)-10*a(n-3)-5*a(n-4)+20*a(n-5)-20*a(n-7)+5*a(n-8)+10*a(n-9)-4*a(n-10)-2*a(n-11)+a(n-12).

G.f.: 2*x^4*(5 + 32*x + 57*x^2 + 146*x^3 + 19*x^4 + 104*x^5 + 15*x^6 + 6*x^7) / ((1 - x)^7*(1 + x)^5). - Colin Barker, Jun 27 2017

Mathematica

Table[(-165 - 282 n + 972 n^2 - 730 n^3 + 280 n^4 - 68 n^5 + 8 n^6 - 5 (-1)^n (-33 - 26 n + 96 n^2 - 46 n^3 + 6 n^4))/240, {n, 20}]
LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {0, 0, 0, 10, 84, 322, 1172, 2780, 7016, 13532, 27720, 47318}, 20]

Prog

(PARI) concat(vector(3), Vec(2*x^4*(5 + 32*x + 57*x^2 + 146*x^3 + 19*x^4 + 104*x^5 + 15*x^6 + 6*x^7) / ((1 - x)^7*(1 + x)^5) + O(x^40))) \\ Colin Barker, Jun 27 2017

Crossrefs

Cf. A289161 (3-cycles), A289162 (4-cycles), A289160 (6-cycles).

Sequence in context: A271557 A351750 A267031 * A092718 A090763 A016131

Adjacent sequences: A289160 A289161 A289162 * A289164 A289165 A289166

Keyword

nonn,easy

Author

Eric W. Weisstein, Jun 26 2017

Status

approved