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

0,3

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

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

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.

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 *)

Prog

(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
(Magma)
C<I>:= 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

Crossrefs

Cf. A000129.

Sequence in context: A047090 A185265 A018178 * A018079 A289920 A060985

Adjacent sequences: A112572 A112573 A112574 * A112576 A112577 A112578

Keyword

easy,nonn

Author

Paul Barry, Sep 14 2005

Status

approved