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 A229026 #16 Feb 24 2025 02:18:03 %S A229026 1,19,238,2490,23631,211509,1823908,15348100,127057261,1040261799, %T A229026 8453319978,68343722910,550640774491,4426107030889,35521389816448, %U A229026 284771933350920,2281370275767321,18267889925254779,146232526369201318,1170331087647336130,9365122293936867751 %N A229026 Expansion of 1/((1-x)*((1-5*x)^2)*(1-8*x)). %C A229026 This sequence was chosen to illustrate a method of solution. %H A229026 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (19,-123,305,-200). %F A229026 a(n) = (2*8^(n+4) - (84*n+287)*5^(n+2) - 9)/1008. %F A229026 In general, for the expansion of 1/((1-t*x)*((1-s*x)^2)*(1-r*x)) with r > s > t, we have the formula: a(n) = (K*r^(n+3) + L*s^(n+3) + M*s^(n+2) + N*t^(n+3))/D, where K, L, M, N, D have the following values: %F A229026 K = (s-t)^2; %F A229026 L = (r-t)*(r-2*s+t); %F A229026 M = -(r-s)*(r-t)*(s-t)*(n+3); %F A229026 N = -(r-s)^2; %F A229026 D = (r-t)*((s-t)^2)*((r-s)^2). %F A229026 Directly using formula we get: a(n) = (16*8^(n+3) - 7*5^(n+3) - 84*(n+3)*5^(n+2) - 9)/1008. After transformation we obtain previous formula. %K A229026 nonn,easy %O A229026 0,2 %A A229026 _Yahia Kahloune_, Sep 18 2013