cp's OEIS Frontend

A242551 Number of n-length words on infinite alphabet {1,2,...} such that the maximal runs of consecutive equal integers have lengths that are at least as great as the integer.

%I A242551 #18 Nov 09 2024 02:42:46
%S A242551 1,1,2,5,11,24,53,118,261,577,1276,2823,6246,13819,30572,67635,149630,
%T A242551 331029,732344,1620187,3584388,7929844,17543415,38811782,85864379,
%U A242551 189960150,420254129,929739922,2056889538,4550514023,10067228909,22272010878,49272989918,109008007822,241161451563,533528195645
%N A242551 Number of n-length words on infinite alphabet {1,2,...} such that the maximal runs of consecutive equal integers have lengths that are at least as great as the integer.
%C A242551 In other words, there is no restriction on the length of runs of 1's, the length of runs of 2's must be at least two, the length of runs of 3's must be at least three...
%C A242551 a(n) is the number of n-color integer compositions of n such that no adjacent parts are the same color. - _John Tyler Rascoe_, Jul 23 2024
%H A242551 Alois P. Heinz, <a href="/A242551/b242551.txt">Table of n, a(n) for n = 0..1000</a>
%F A242551 G.f.: 1/(1 - Sum_{i>0} x^i/(1 - x + x^i)). - _John Tyler Rascoe_, Jul 23 2024
%e A242551 a(3)=5 because we have: 111, 122, 221, 222, 333.
%e A242551 a(4)=11 because we have:  1111, 1122, 1221, 1222, 2211, 2221, 2222, 3331, 1333, 3333, 4444.
%p A242551 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A242551       `if`(t=0, 0, b(n-1, t)) +add(
%p A242551       `if`(t=j, 0, b(n-j, j)), j=1..n))
%p A242551     end:
%p A242551 a:= n-> b(n, 0):
%p A242551 seq(a(n), n=0..40);  # _Alois P. Heinz_, Oct 07 2015
%t A242551 n=nn=35;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->z^i/(1-z),{i,1,n}]),{z,0,nn}],z]
%o A242551 (PARI)
%o A242551 C_x(N)={my(x='x+O('x^N), h = 1/(1-sum(i=1,N, x^i/(1 - x + x^i)))); Vec(h)}
%o A242551 C_x(40) \\ _John Tyler Rascoe_, Jul 23 2024
%Y A242551 Cf. A088305, A351638.
%K A242551 nonn,easy
%O A242551 0,3
%A A242551 _Geoffrey Critzer_, May 17 2014