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 A385329 #17 Jul 05 2025 09:59:54
%S A385329 0,0,0,0,6,110,1220,10612,79786,544434,3468792,21012200,122500334,
%T A385329 693324502,3833742796,20809676604,111288341970,588046458074,
%U A385329 3076991784512,15972440574064,82370489136214,422506631928510,2157589903432020,10977781519321220,55686118748465786
%N A385329 a(n) = 5^n - 2*4^(n-1)*(n+4) + 3^(n-2)*(n^2+5*n+9).
%C A385329 a(n) is the number of words of length n defined on 5 letters where two chosen letters, say a and b, are used at least twice.
%H A385329 Vincenzo Librandi, <a href="/A385329/b385329.txt">Table of n, a(n) for n = 0..500</a>
%H A385329 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (22,-200,962,-2583,3672,-2160).
%F A385329 E.g.f.: exp(3*x)*(exp(x) - x - 1)^2.
%F A385329 G.f.: 2*x^4*(3 - 11*x)/((1 - 4*x)^2*(1 - 3*x)^3*(1 - 5*x)). - _Jinyuan Wang_, Jun 26 2025
%e A385329 a(4) = 6 since the words are the 6 permutations of aabb.
%e A385329 a(5) = 110 since the words are (number of permutations in parentheses): aaabb (10), aabbb (10), aabbc (30), aabbd (30), aabbe (30).
%t A385329 LinearRecurrence[{22, -200, 962, -2583, 3672, -2160}, {0, 0, 0, 0, 6, 110, 1220}, 25] (* _Amiram Eldar_, Jun 28 2025 *)
%o A385329 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0,0,0,0] cat Coefficients(R!((2*x^4*(3 - 11*x)/((1 - 4*x)^2*(1 - 3*x)^3*(1 - 5*x))))); // _Vincenzo Librandi_, Jul 05 2025
%Y A385329 Cf. A112495, A381646.
%K A385329 nonn,easy
%O A385329 0,5
%A A385329 _Enrique Navarrete_, Jun 25 2025