login
A112577
A Chebyshev-related transform of the Jacobsthal numbers.
3
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)).
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 14 2005
STATUS
approved