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 A141416 #13 Sep 04 2024 15:30:46
%S A141416 -1,-1,2,0,0,-2,4,-2,4,-6,12,-10,20,-22,44,-42,84,-86,172,-170,340,
%T A141416 -342,684,-682,1364,-1366,2732,-2730,5460,-5462,10924,-10922,21844,
%U A141416 -21846,43692,-43690,87380,-87382,174764,-174762,349524,-349526,699052,-699050,1398100,-1398102
%N A141416 First differences of A133730.
%H A141416 G. C. Greubel, <a href="/A141416/b141416.txt">Table of n, a(n) for n = 0..1000</a>
%H A141416 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2).
%F A141416 a(2n) = (-1)^(n+1)*A084247(n).
%F A141416 a(2n+1) = -A078008(n).
%F A141416 a(2n) = -2*a(2n-1), n>0.
%F A141416 a(2n) + a(2n+1) = 2*(-1)^(n+1).
%F A141416 G.f.: (-1 -x +3*x^2 +x^3)/( (1-2*x^2)*(1+x^2) ). - _R. J. Mathar_, Jul 02 2011
%F A141416 a(n) = ((4*i^(n+1) - 2^((n+1)/2))*(1-(-1)^n) - 2*(4*i^n - 2^(n/2))*(1+(-1)^n))/12. - _G. C. Greubel_, Mar 30 2021
%t A141416 LinearRecurrence[{0,1,0,2}, {-1,-1,2,0}, 50] (* _G. C. Greubel_, Mar 30 2021 *)
%t A141416 Differences[LinearRecurrence[{0,1,0,2},{1,0,-1,1},70]] (* _Harvey P. Dale_, Sep 04 2024 *)
%o A141416 (Magma) I:=[-1,-1,2,0]; [n le 4 select I[n] else Self(n-2) +2*Self(n-4): n in [1..51]]; // _G. C. Greubel_, Mar 30 2021
%o A141416 (Sage) [((4*i^(n+1) - 2^((n+1)/2))*(1-(-1)^n) - 2*(4*i^n - 2^(n/2))*(1+(-1)^n))/12 for n in (0..50)] # _G. C. Greubel_, Mar 30 2021
%Y A141416 Cf. A078008, A084247.
%K A141416 sign,easy
%O A141416 0,3
%A A141416 _Paul Curtz_, Aug 05 2008