A112577 A Chebyshev-related transform of the Jacobsthal numbers.

0, 1, 1, 5, 8, 26, 52, 143, 317, 811, 1884, 4668, 11076, 27053, 64805, 157273, 378364, 915598, 2206976, 5333731, 12867673, 31080023, 75010008, 181128696, 437221032, 1055645785, 2548391209, 6152624621, 14853322640, 35859784130, 86572058860

Offset

0,4

Comments

Transform of the Jacobsthal numbers by the Chebyshev related transform which maps g(x) -> (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 (1,4,-1,-1).

Formula

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A001045(n-2*k).

a(n) = (1/2)*Sum_{k=0..n} binomial((n+k)/2, k)*(1 + (-1)^(n-k))*A001045(k).

a(n) = Sum_{k=0..n} (-1)^k*Fibonacci(k+1)*A000129(n-k).

a(n) = (A000129(n+1) - A039834(n+1))/3. - R. J. Mathar, Sep 20 2012

Mathematica

LinearRecurrence[{1, 4, -1, -1}, {0, 1, 1, 5}, 40] (* G. C. Greubel, Jan 14 2022 *)

Prog

(Sage) [sum(binomial(n-k, k)*lucas_number1(n-2*k, 1, -2) for k in (0..(n/2))) for n in (0..40)] # G. C. Greubel, Jan 14 2022
(Magma)
J:= func< n | (2^n - (-1)^n)/3 >; // A001045
[(&+[Binomial(n-k, k)*J(n-2*k): k in [0..Floor(n/2)]]) : n in [0..40]]; // _G. C. Greubel, Jan 14 2022

Crossrefs

Cf. A000045, A000129, A001045, A039834.

Sequence in context: A038250 A067973 A101584 * A192920 A162562 A182543

Adjacent sequences: A112574 A112575 A112576 * A112578 A112579 A112580

Keyword

easy,nonn

Author

Paul Barry, Sep 14 2005

Status

approved