Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Jan 15 2022 09:47:59
%S 0,1,1,5,8,26,52,143,317,811,1884,4668,11076,27053,64805,157273,
%T 378364,915598,2206976,5333731,12867673,31080023,75010008,181128696,
%U 437221032,1055645785,2548391209,6152624621,14853322640,35859784130,86572058860
%N A Chebyshev-related transform of the Jacobsthal numbers.
%C Transform of the Jacobsthal numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))*g(x/(1-x^2)).
%H G. C. Greubel, <a href="/A112577/b112577.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-1).
%F G.f.: x/( (1+x-x^2)*(1-2*x-x^2) ).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A001045(n-2*k).
%F a(n) = (1/2)*Sum_{k=0..n} binomial((n+k)/2, k)*(1 + (-1)^(n-k))*A001045(k).
%F a(n) = Sum_{k=0..n} (-1)^k*Fibonacci(k+1)*A000129(n-k).
%F a(n) = (A000129(n+1) - A039834(n+1))/3. - _R. J. Mathar_, Sep 20 2012
%t LinearRecurrence[{1,4,-1,-1}, {0,1,1,5}, 40] (* _G. C. Greubel_, Jan 14 2022 *)
%o (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
%o (Magma)
%o J:= func< n | (2^n - (-1)^n)/3 >; // A001045
%o [(&+[Binomial(n-k,k)*J(n-2*k): k in [0..Floor(n/2)]]) : n in [0..40]]; // _G. C. Greubel, Jan 14 2022
%Y Cf. A000045, A000129, A001045, A039834.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Sep 14 2005