A112575 Chebyshev transform of the second kind of the Pell numbers.

0, 1, 2, 3, 6, 12, 22, 41, 78, 147, 276, 520, 980, 1845, 3474, 6543, 12322, 23204, 43698, 82293, 154974, 291847, 549608, 1035024, 1949160, 3670665, 6912610, 13017851, 24515262, 46167228, 86942286, 163730017, 308336942, 580661211, 1093503228, 2059289112

Offset: 0

Author

Paul Barry, Sep 14 2005

Keywords

• easy
• nonn

Comments

The Chebyshev transform of the second kind maps the sequence with g.f. g(x) to the sequence with g.f. (1/(1+x^2))g(x/(1+x^2)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See pp. 9-10.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-1).

Crossrefs

Cf. A000129.

Programs

Magma

C:= ComplexField();
[(&+[Binomial(n-k,k)*Round(I^(n-1)*Evaluate(ChebyshevU(n-2*k), -I)): k in [0..Floor(n/2)]]) : n in [0..40]]; // G. C. Greubel, Jan 14 2022

Mathematica

Table[Sum[(-1)^k*Binomial[n-k, k]*Fibonacci[n-2*k, 2], {k,0,Floor[n/2]}], {n, 0, 40}] (* G. C. Greubel, Jan 14 2022 *)

Sage

[sum((-1)^k*binomial(n-k,k)*lucas_number1(n-2*k, 2, -1) for k in (0..(n/2))) for n in (0..40)] # G. C. Greubel, Jan 14 2022

Formula

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

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k, k)*A000129(n-2k).

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