A391027 Number of tetrahedra in the n X n white bishop graph.

0, 0, 0, 1, 4, 27, 64, 182, 344, 714, 1184, 2079, 3164, 5005, 7168, 10556, 14448, 20196, 26688, 35853, 46068, 59983, 75328, 95634, 117832, 146510, 177632, 217035, 259532, 312417, 369152, 438712, 512992, 602888, 698496, 812889, 934116, 1077699, 1229376, 1407406

Offset

1,5

Comments

Sequence extended to a(1) using the formula.

Links

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

Eric Weisstein's World of Mathematics, Graph Tetrahedron.

Eric Weisstein's World of Mathematics, White Bishop Graph.

Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Formula

a(n) = (n^2 - 4*n + 3)*(4*n^3 - 14*n^2 + 2*n + 5 + 5*(-1)^n*(2*n - 1))/240.

G.f.: x^4*(1+2*x+16*x^2+6*x^3+7*x^4)/((-1+x)^6*(1+x)^4).

a(n) = 2*a(n-1)+3*a(n-2)-8*a(n-3)-2*a(n-4)+12*a(n-5)-2*a(n-6)-8*a(n-7)+3*a(n-8)+2*a(n-9)-a(n-10).

Mathematica

Table[(n^2 - 4 n + 3) (4 n^3 - 14 n^2 + 2 n + 5 + 5 (-1)^n (2 n - 1))/240, {n, 20}]
LinearRecurrence[{2, 3, -8, -2, 12, -2, -8, 3, 2, -1}, {0, 0, 0, 1, 4, 27, 64, 182, 344, 714}, 20]
CoefficientList[Series[x^3 (1 + 2 x + 16 x^2 + 6 x^3 + 7 x^4)/((x - 1)^6 (x + 1)^4), {x, 0, 20}], x]

Crossrefs

Sequence in context: A349200 A078111 A186882 * A097792 A308474 A058067

Adjacent sequences: A391024 A391025 A391026 * A391028 A391029 A391030

Keyword

nonn,easy

Author

Eric W. Weisstein, Jan 12 2026

Status

approved