A166989 G.f.: A(x) = 1/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).

1, 2, 11, 38, 156, 598, 2353, 9166, 35843, 139956, 546792, 2135796, 8343205, 32590610, 127308455, 497301794, 1942600788, 7588340434, 29642181517, 115790645854, 452310642407, 1766851828392, 6901817263824, 26960427965352

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Formula

G.f.: A(x) = exp( Sum_{n>=1} A000204(n)*A002203(n)*x^n/n ) where A000204 (Lucas numbers) forms the logarithmic derivative of the Fibonacci numbers (A000045) and A002203 forms the logarithmic derivative of the Pell numbers (A000129).

Recurrence: a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4) where a(k)=0 for k<0 with a(0)=1.

Radius of convergence: r = f*p where f=(sqrt(5)-1)/2, p=sqrt(2)-1:

(f*p-x)*(1/(f*p)-x)*(f/p+x)*(p/f+x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4.

For n >= 2, a(n) - a(n-2) = Fibonacci(n+1)*Pell(n+1) = A001582(n). - Peter Bala, Aug 30 2015

a(n) = (1/2) * Sum_{i=0..n+2} (-1)^(n-i+1) * Lucas(2*i-n-2) * Pell(i) * Pell(n+2-i). - Vladimir Kruchinin, Jan 10 2025

Mathematica

LinearRecurrence[{2, 7, 2, -1}, {1, 2, 11, 38}, 100] (* G. C. Greubel, May 30 2016 *)

Prog

(PARI) {a(n)=polcoeff(1/(1-2*x-7*x^2-2*x^3+x^4+x*O(x^n)), n)}
(PARI) {a(n)=if(n<0, 0, if(n==0, 1, 2*a(n-1)+7*a(n-2)+2*a(n-3)-a(n-4)))}

Crossrefs

Cf. A000204, A000045, A002203, A000129, A001582.

Sequence in context: A196701 A196850 A203534 * A143550 A259213 A259658

Adjacent sequences: A166986 A166987 A166988 * A166990 A166991 A166992

Keyword

nonn

Author

Paul D. Hanna, Oct 26 2009

Status

approved