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 A191272 #33 Sep 08 2022 08:45:57
%S A191272 0,4,17,63,256,1025,4095,16384,65537,262143,1048576,4194305,16777215,
%T A191272 67108864,268435457,1073741823,4294967296,17179869185,68719476735,
%U A191272 274877906944,1099511627777,4398046511103,17592186044416,70368744177665
%N A191272 Expansion of x*(4+5*x)/( (1-4*x)*(1 + x + x^2) ).
%H A191272 Vincenzo Librandi, <a href="/A191272/b191272.txt">Table of n, a(n) for n = 0..175</a>
%H A191272 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,4).
%F A191272 G.f.: x*(4+5*x)/(1 - 3*x - 3*x^2 - 4*x^3).
%F A191272 a(n) = 4^n-A057078(n) = 4^n - (n-th element of periodic length 3 repeat 1, 0, -1)
%F A191272 a(n) = A024495(2*n) + A024495(1+2*n).
%F A191272 a(n+1) = 4*a(n) + (n-th element of periodic length 3 repeat 4, 1, -5).
%F A191272 a(n) = A052539(n) - A080425(n+1). - _Bruno Berselli_, Jun 06 2011
%F A191272 a(0)=0, a(1)=4, a(2)=17, a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3). - _Harvey P. Dale_, Jun 19 2011
%F A191272 a(n) = 4*a(n-1) + a(n-3) - 4*(n-4) (n>3). - _Bruno Berselli_, Jul 04 2011
%t A191272 LinearRecurrence[{3,3,4},{0,4,17},30] (* or *) CoefficientList[ Series[ x (4+5x)/((1-4x)(1+x+x^2)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 19 2011 *)
%o A191272 (PARI) a(n)=4^n-[1,0,-1][n%3+1] \\ _Charles R Greathouse IV_, Jun 06 2011
%o A191272 (Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(4+5*x)/((1-4*x)*(1+x+x^2)))); // _Bruno Berselli_, Jul 04 2011
%o A191272 (Maxima) makelist(coeff(taylor(x*(4+5*x)/((1-4*x)*(1+x+x^2)), x, 0, n), x, n), n, 0, 23); /* _Bruno Berselli_, Jun 06 2011 */
%K A191272 nonn,easy
%O A191272 0,2
%A A191272 _Paul Curtz_, May 29 2011