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 A191008 #23 Jul 31 2021 03:27:53
%S A191008 1,5,22,86,319,1139,3964,13532,45517,151313,498226,1627538,5281195,
%T A191008 17039327,54705208,174877304,556916953,1767605981,5593383310,
%U A191008 17651846030,55570626391,174557144075,547207226932,1712229064916,5348509347109,16680994498409,51949382866474
%N A191008 a(n) = (n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4.
%C A191008 Another renewal type of sequence. Let X, X(1), X(2),... denote random variables with pdf P(X = 1) = P(X = 4 ) = 1/4 and P(X = 2) = 1/2. Let N(x) denote the first value of k such that X(1)*X(2)*...*X(k) > x and let H(x)= E(N(x)). The sequence is given by a(n) = 3^(n+1)*H(2^n)/4.
%H A191008 Colin Barker, <a href="/A191008/b191008.txt">Table of n, a(n) for n = 0..1000</a>
%H A191008 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-9).
%F A191008 a(n) = (n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4.
%F A191008 From _Colin Barker_, May 03 2017: (Start)
%F A191008 G.f.: 1 / ((1 + x)*(1 - 3*x)^2).
%F A191008 a(n) = 5*a(n-1) - 3*a(n-2) - 9*a(n-3) for n>2.
%F A191008 (End)
%p A191008 A191008:=n->(n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4: seq(A191008(n), n=0..40); # _Wesley Ivan Hurt_, May 03 2017
%t A191008 LinearRecurrence[{5, -3, -9}, {1, 5, 22}, 27] (* or *)
%t A191008 CoefficientList[Series[1/((1 + x) (1 - 3 x)^2), {x, 0, 26}], x] (* _Michael De Vlieger_, May 03 2017 *)
%o A191008 (PARI) a(n)=(n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4; \\ _Michel Marcus_, Oct 16 2014
%o A191008 (PARI) Vec(1 / ((1 + x)*(1 - 3*x)^2) + O(x^30)) \\ _Colin Barker_, May 03 2017
%K A191008 nonn,easy
%O A191008 0,2
%A A191008 _Edward Omey_, Jun 16 2011
%E A191008 More terms from _Michel Marcus_, Oct 16 2014