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 A100088 #25 Apr 22 2025 12:16:33
%S A100088 1,2,2,4,10,20,38,76,154,308,614,1228,2458,4916,9830,19660,39322,
%T A100088 78644,157286,314572,629146,1258292,2516582,5033164,10066330,20132660,
%U A100088 40265318,80530636,161061274,322122548,644245094,1288490188,2576980378
%N A100088 Expansion of (1-x^2)/((1-2*x)*(1+x^2)).
%C A100088 A Chebyshev transform of A100087, under the mapping A(x) -> ((1-x^2)/(1+x^2)) * A(x/(1+x^2)).
%C A100088 A176742(n+2) = A084099(n+2) = period 4:repeat 0, -2, 0, 2.
%H A100088 G. C. Greubel, <a href="/A100088/b100088.txt">Table of n, a(n) for n = 0..1000</a>
%H A100088 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2).
%F A100088 a(n) = (3*2^n + 2*cos(Pi*n/2) + 4*sin(Pi*n/2))/5.
%F A100088 a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A100087(n-2*k)/(n-k).
%F A100088 a(n) = 2*a(n-1) + period 4:repeat 0, -2, 0, 2, with a(0) = 1.
%F A100088 a(n) = A007910(n+1) - A007910(n-1).
%F A100088 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
%F A100088 a(n) = (1/5)*(3*2^n + i^n*(1+(-1)^n) - 2*i^(n+1)*(1-(-1)^n)). - _G. C. Greubel_, Jul 08 2022
%F A100088 a(n) = A122117(n/2) if (n mod 2 = 0) otherwise 2*A122117((n-1)/2). - _G. C. Greubel_, Jul 21 2022
%t A100088 CoefficientList[Series[(1-x^2)/((1-2x)(1+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2},{1,2,2},40] (* _Harvey P. Dale_, May 12 2011 *)
%o A100088 (Magma) [n le 3 select Floor((n+2)/2) else 2*Self(n-1) - Self(n-2) +2*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Jul 08 2022
%o A100088 (SageMath)
%o A100088 def b(n): return (2/5)*(3*2^(2*n-1) + (-1)^n) # b=A122117
%o A100088 def A100088(n): return b(n/2) if (n%2==0) else 2*b((n-1)/2)
%o A100088 [A100088(n) for n in (0..60)] # _G. C. Greubel_, Jul 08 2022
%o A100088 (Python)
%o A100088 def A100088(n): return ((4<<n)|2)//5-((1<<n)|2)//5 # _Chai Wah Wu_, Apr 22 2025
%Y A100088 Cf. A007910, A084099, A100087, A122117, A176742.
%K A100088 easy,nonn
%O A100088 0,2
%A A100088 _Paul Barry_, Nov 03 2004