A080872 a(n)*a(n+3) - a(n+1)*a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

1, 1, 5, 9, 49, 89, 485, 881, 4801, 8721, 47525, 86329, 470449, 854569, 4656965, 8459361, 46099201, 83739041, 456335045, 828931049, 4517251249, 8205571449, 44716177445, 81226783441, 442644523201, 804062262961, 4381729054565, 7959395846169, 43374646022449, 78789896198729, 429364731169925

Offset: 0

Author

Paul D. Hanna, Feb 22 2003

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Crossrefs

Cf. A080871, A080873, A080874, A080875.

Bisections are A001079 and A072256.

Programs

Mathematica

CoefficientList[Series[(-x^3-5 x^2+x+1)/(x^4-10 x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,5,9},30] (* Harvey P. Dale, May 06 2012 *)

PARI

Vec( (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1) + O(x^66) ) \\ Joerg Arndt, Jan 29 2016

Formula

G.f.: (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1).

a(n) = (3+sqrt(3))/12*(sqrt(3)-sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)+sqrt(2))^n+(3+sqrt(3))/12*(sqrt(3)+sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)-sqrt(2))^n. [Richard Choulet, Dec 03 2008]

a(n+4) = 10*a(n+2)-a(n). [Richard Choulet, Dec 04 2008]