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 A112575 #30 May 18 2025 04:08:52
%S A112575 0,1,2,3,6,12,22,41,78,147,276,520,980,1845,3474,6543,12322,23204,
%T A112575 43698,82293,154974,291847,549608,1035024,1949160,3670665,6912610,
%U A112575 13017851,24515262,46167228,86942286,163730017,308336942,580661211,1093503228,2059289112
%N A112575 Chebyshev transform of the second kind of the Pell numbers.
%C A112575 The Chebyshev transform of the second kind maps the sequence with g.f. g(x) to the sequence with g.f. (1/(1+x^2))g(x/(1+x^2)).
%H A112575 G. C. Greubel, <a href="/A112575/b112575.txt">Table of n, a(n) for n = 0..1000</a>
%H A112575 Jia Huang, <a href="https://arxiv.org/abs/2501.07463">A coin flip game and generalizations of Fibonacci numbers</a>, arXiv:2501.07463 [math.CO], 2025. See pp. 9-10.
%H A112575 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-1).
%F A112575 G.f.: x/(1-2*x+x^2-2*x^3+x^4).
%F A112575 a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k, k)*A000129(n-2k).
%F A112575 a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*C((n+k)/2, k)*(1+(-1)^(n-k))*A000129(k)/2.
%t A112575 Table[Sum[(-1)^k*Binomial[n-k, k]*Fibonacci[n-2*k, 2], {k,0,Floor[n/2]}], {n, 0, 40}] (* _G. C. Greubel_, Jan 14 2022 *)
%o A112575 (Sage) [sum((-1)^k*binomial(n-k,k)*lucas_number1(n-2*k, 2, -1) for k in (0..(n/2))) for n in (0..40)] # _G. C. Greubel_, Jan 14 2022
%o A112575 (Magma)
%o A112575 C<I>:= ComplexField();
%o A112575 [(&+[Binomial(n-k,k)*Round(I^(n-1)*Evaluate(ChebyshevU(n-2*k), -I)): k in [0..Floor(n/2)]]) : n in [0..40]]; // _G. C. Greubel_, Jan 14 2022
%Y A112575 Cf. A000129.
%K A112575 easy,nonn
%O A112575 0,3
%A A112575 _Paul Barry_, Sep 14 2005