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 A191697 #33 Sep 08 2022 08:45:57
%S A191697 0,4,6,8,20,28,56,96,168,304,528,944,1664,2944,5216,9216,16320,28864,
%T A191697 51072,90368,159872,282880,500480,885504,1566720,2771968,4904448,
%U A191697 8677376,15352832,27163648,48060416,85032960,150448128,266186752,470962176,833269760
%N A191697 a(n) = r1^n + r2^n + r3^n where r1, r2, r3 are the three roots of x^3 - 2*x - 2 = 0.
%C A191697 Definition 1.1 defines F_3^n as a Boolean function and definition 1.3 defines the Fourier transform of a Boolean function. - _Michael Somos_, Aug 04 2012
%H A191697 G. C. Greubel, <a href="/A191697/b191697.txt">Table of n, a(n) for n = 1..2500</a>
%H A191697 X. Zhang, H. Guo, R. Feng and Y. Li, <a href="https://doi.org/10.1016/j.disc.2011.03.012">Proof of a conjecture about rotation symmetric functions</a>, Discrete Math., 311 (2011), 1281-1289.
%H A191697 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2).
%F A191697 a(n) = hat{F_3^n}(0), the Fourier transform evaluated at 0 of the Boolean function F_3^n defined by F_3^n(x_0, ..., x_{n-1}) = Sum_{ 0<i<=n-1} x_i x_{i+1(mod n)} x_{i+2(mod n)}. - _Michael Somos_, Aug 04 2012
%F A191697 From _Michael Somos_, Aug 04 2012: (Start)
%F A191697 G.f.: 2 * x^2 * (2 + 3*x) / (1 - 2*x^2 - 2*x^3).
%F A191697 a(n + 3) = 2*a(n + 1) + 2*a(n). (End)
%F A191697 a(n) = 4*A052907(n) +6*A052907(n-1). - _R. J. Mathar_, Aug 10 2012
%e A191697 G.f. = 4*x^2 + 6*x^3 + 8*x^4 + 20*x^5 + 28*x^6 + 56*x^7 + 96*x^8 + 168*x^9 + ...
%t A191697 Rest[CoefficientList[Series[2*x^2*(2+3*x)/(1-2*x^2-2*x^3), {x, 0, 50}], x]] (* _G. C. Greubel_, Aug 13 2018 *)
%t A191697 LinearRecurrence[{0, 2, 2}, {0, 4, 6}, 40] (* _Vincenzo Librandi_, Aug 14 2018 *)
%o A191697 (PARI) {a(n) = if( n<1, 0, polsym( x^3 - 2*x - 2, n)[n + 1])}; /* _Michael Somos_, Aug 04 2012 */
%o A191697 (PARI) {a(n) = if( n<1, 0, sum( x=0, 2^n-1, (-1)^sum( i=0, n-1, bittest(x, i) * bittest(x, (i+1)%n) * bittest(x, (i+2)%n))))}; /* _Michael Somos_, Aug 04 2012 */
%o A191697 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x^2*(2+3*x)/(1-2*x^2-2*x^3))); // _G. C. Greubel_, Aug 13 2018
%o A191697 (Magma) I:=[0,4,6]; [n le 3 select I[n] else 2*Self(n-2)+2*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Aug 14 2018
%K A191697 nonn,easy
%O A191697 1,2
%A A191697 _N. J. A. Sloane_, Jun 19 2011