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%I A351643 #10 Jan 28 2023 22:07:47
%S A351643 0,0,1,3,12,28,81,177,410,906,1869,4001,8094,16032,32355,62499,120078,
%T A351643 227880,436743,805797,1487920,2751618,5017143,9063625,16153560,
%U A351643 29066676,51334289,90784671,157941132,275244344,478874505,823848357,1412686722,2400778830,4091929101
%N A351643 Number of length n word structures with all distinct runs using exactly 3 symbols.
%C A351643 Permuting the symbols will not change the structure.
%H A351643 Andrew Howroyd, <a href="/A351643/b351643.txt">Table of n, a(n) for n = 1..1000</a>
%e A351643 The a(3) = 1 word is 123.
%e A351643 The a(4) = 3 words are 1123, 1223, 1233.
%e A351643 The a(5) = 12 words are 11123, 11213, 11223, 11231, 11233, 12113, 12223, 12232, 12233, 12311, 12322, 12333.
%o A351643 (PARI) \\ See A351641 for R, S.
%o A351643 seq(n)={my(q=S(n), c=3); sum(k=1, c, R(q^k-1)*binomial(c, k)*(-1)^(3-k))/c!}
%Y A351643 Column k=3 of A351641.
%Y A351643 Cf. A351644.
%K A351643 nonn
%O A351643 1,4
%A A351643 _Andrew Howroyd_, Feb 16 2022

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