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

Author

Paul Barry, Sep 14 2005

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000045, A000129, A001045, A039834.

Programs

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

Mathematica

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

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

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