A101893 a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*Pell(k).

0, 0, 1, 3, 8, 20, 50, 126, 320, 816, 2084, 5324, 13600, 34736, 88712, 226552, 578560, 1477504, 3773200, 9635888, 24607872, 62842944, 160486688, 409846752, 1046656000, 2672922880, 6826040896, 17432165568, 44517810688, 113688426240

Offset: 0

Author

Paul Barry, Dec 22 2004

Keywords

• easy
• nonn

Comments

Transform of Pell numbers under the mapping g(x)-> (1/(1-x))*g(x^2/((1-x)^2)).

Binomial transform of aerated Pell numbers 0,0,1,0,2,0,5,0,12,...

Links

• Andrew Woods, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,2).

Crossrefs

Cf. A000129 (Pell numbers), A135248 (partial sums).

Programs

Mathematica

CoefficientList[Series[x^2*(1-x)/(1 - 4*x + 4*x^2 - 2*x^4), {x, 0, 40}], x] (* Vaclav Kotesovec, Jan 05 2015 *)
LinearRecurrence[{4,-4,0,2},{0,0,1,3},30] (* Harvey P. Dale, Aug 05 2018 *)

Formula

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

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-4).

a(n) = sum_{k=0..n} binomial(n, k) * A000129(k/2) * (1+(-1)^k)/2.