A120711 Expansion of 2*x*(7+16*x-2*x^2-14*x^3)/(1-11*x^2-12*x^3+10*x^4+12*x^5).

0, 14, 32, 150, 492, 1894, 6724, 24854, 89972, 329238, 1197972, 4372054, 15930580, 58096214, 211770452, 772129110, 2814859092, 10262536534, 37414140244, 136403674454, 497291840852, 1813006427478, 6609762501972, 24097566365014

Offset: 0

Author

Roger L. Bagula, Aug 12 2006

Keywords

• nonn
• easy

Comments

Former title: 7 X 7 matrix Matrov of seven vertex Fano Plane: Characteristic polynomial: 12 + 10*x - 24*x^2 - 21*x^3 + 12*x^4 + 12*x^5 - x^7.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Fano Plane
• Index entries for linear recurrences with constant coefficients, signature (0,11,12,-10,-12).

Crossrefs

Cf. A083099, A111384.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 2*x*(7+16*x-2*x^2-14*x^3)/(1-11*x^2-12*x^3+10*x^4+12*x^5) )); // G. C. Greubel, Jul 22 2023

Mathematica

M = {{0,1,0,0,0,1,1}, {1,0,1,0,0,0,1}, {0,1,0,1,0,0,1}, {0,0,1,0,1,0, 1}, {0,0,0,1,0,1,1}, {1,0,0,0,1,0,1}, {1,1,1,1,1,1,0}};
v[1] = {0,1,1,2,3,5,8}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n,50}]
LinearRecurrence[{0,11,12,-10,-12}, {0,14,32,150,492}, 40] (* G. C. Greubel, Jul 22 2023 *)

SageMath

A083099=BinaryRecurrenceSequence(2,6,0,1)
def A120711(n): return (1/3)*(-1 -3*(-1)^n +(-2)^(n+1) +6*(A083099(n+1) +4*A083099(n)))
[A120711(n) for n in range(41)] # G. C. Greubel, Jul 22 2023

Formula

a(n) = 11*a(n-2) + 12*a(n-3) - 10*a(n-4) - 12*a(n-5).

G.f.: 2*x*(7+16*x-2*x^2-14*x^3)/((1-x)*(1+x)*(1+2*x)*(1-2*x-6*x^2)). - Colin Barker, Mar 26 2012

a(n) = (1/3)*(-1 - 3*(-1)^n + (-2)^(n+1) + 6*(A083099(n+1) + 4*A083099(n))). - G. C. Greubel, Jul 22 2023

Extensions

Edited by G. C. Greubel, Jul 22 2023