This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A112577 #10 Jan 15 2022 09:47:59
%S A112577 0,1,1,5,8,26,52,143,317,811,1884,4668,11076,27053,64805,157273,
%T A112577 378364,915598,2206976,5333731,12867673,31080023,75010008,181128696,
%U A112577 437221032,1055645785,2548391209,6152624621,14853322640,35859784130,86572058860
%N A112577 A Chebyshev-related transform of the Jacobsthal numbers.
%C A112577 Transform of the Jacobsthal numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))*g(x/(1-x^2)).
%H A112577 G. C. Greubel, <a href="/A112577/b112577.txt">Table of n, a(n) for n = 0..1000</a>
%H A112577 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-1).
%F A112577 G.f.: x/( (1+x-x^2)*(1-2*x-x^2) ).
%F A112577 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A001045(n-2*k).
%F A112577 a(n) = (1/2)*Sum_{k=0..n} binomial((n+k)/2, k)*(1 + (-1)^(n-k))*A001045(k).
%F A112577 a(n) = Sum_{k=0..n} (-1)^k*Fibonacci(k+1)*A000129(n-k).
%F A112577 a(n) = (A000129(n+1) - A039834(n+1))/3. - _R. J. Mathar_, Sep 20 2012
%t A112577 LinearRecurrence[{1,4,-1,-1}, {0,1,1,5}, 40] (* _G. C. Greubel_, Jan 14 2022 *)
%o A112577 (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 A112577 (Magma)
%o A112577 J:= func< n | (2^n - (-1)^n)/3 >; // A001045
%o A112577 [(&+[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 A112577 Cf. A000045, A000129, A001045, A039834.
%K A112577 easy,nonn
%O A112577 0,4
%A A112577 _Paul Barry_, Sep 14 2005