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 A143462 #21 Oct 02 2024 01:58:07
%S A143462 1,-4,8,0,-64,256,-512,0,4096,-16384,32768,0,-262144,1048576,-2097152,
%T A143462 0,16777216,-67108864,134217728,0,-1073741824,4294967296,-8589934592,
%U A143462 0,68719476736,-274877906944,549755813888,0,-4398046511104,17592186044416
%N A143462 Expansion of 1/(1 + 4*x + 8*x^2).
%H A143462 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-8).
%F A143462 G.f.: 1/(1 + 4*x + 8*x^2).
%F A143462 E.g.f.: (cos(2*x) - sin(2*x))/exp(2*x).
%F A143462 a(n) = -4*a(n-1) - 8*a(n-2).
%F A143462 a(n+4) = -64*a(n).
%F A143462 G.f.: 1/(1 + 4*x/(1 - 2*x/(1 + 2*x))) = 1 - 4*x/(1 + 2*x/(1 - 2*x/(1 + 4*x))). - _Michael Somos_, Jan 03 2013
%F A143462 a(n) = (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2) for n >= 1. - _Peter Luschny_, Dec 17 2015
%e A143462 1 - 4*x + 8*x^2 - 64*x^4 + 256*x^5 - 512*x^6 + 4096*x^8 - 16384*x^9 + ...
%p A143462 A143462 := n -> `if`(n=0, 1, (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):
%p A143462 seq(simplify(A143462(n)), n=0..29); # _Peter Luschny_, Dec 17 2015
%t A143462 CoefficientList[Series[1/(1 + 4*x + 8*x^2), {x, 0, 30}], x] (* _Jinyuan Wang_, Mar 10 2020 *)
%o A143462 (PARI) {a(n) = (-64)^(n \ 4) * [1, -4, 8, 0][n%4 + 1]}
%o A143462 (PARI) {a(n) = n--; -2 * 2^n * ((-1 + I)^n + (-1 - I)^n)}
%o A143462 (PARI) {a(n) = n--; simplify( -4 * (2 * quadgen(8))^n * polchebyshev(n, 1, -1 / quadgen(8)))}
%o A143462 (Magma) I:=[1,-4]; [n le 2 select I[n] else -4 * Self(n-1) - 8 * Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 17 2015
%Y A143462 A030210(2^n) = 2^n * A108520(n) = a(n).
%K A143462 sign,easy
%O A143462 0,2
%A A143462 _Michael Somos_, Aug 16 2008