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 A384162 #8 May 27 2025 17:55:58
%S A384162 1,6,45,460,5945,92736,1694329,35487432,838341009,22054058290,
%T A384162 639434542021,20260243575936,696512594466793,25822887652517970,
%U A384162 1027054229302256625,43622499402922710256,1970666970910292873249,94353519890358073478880,4772755056209685781141981
%N A384162 Number of length n words over an n-ary alphabet such that a single letter in every run of letters is marked.
%F A384162 a(n) = [x^n] n*x/(1 - x*(1 - x + n)).
%F A384162 a(n) = Sum_{s} Product_{i=1..k} c_i * (n - 1 + [i,1]) where the sum is over all compositions of n, [c_1, c_2, ..., c_k].
%F A384162 Conjecture: a(n) = n * A342168(n-1).
%e A384162 a(2) = 6 counts: (1#,1), (1,1#), (1#,2#), (2#,1#), (2#,2), (2,2#) where # denotes a mark.
%o A384162 (PARI)
%o A384162 a(n) = concat([0],Vec(n*x/(1-x*(1-x+n))+O('x^(n+1))))[n+1]
%Y A384162 Cf. A000312, A011782, A342168, A351016, A351638.
%K A384162 nonn,easy
%O A384162 1,2
%A A384162 _John Tyler Rascoe_, May 21 2025