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%I #11 Jul 19 2024 14:35:29
%S 1,1,3,2,4,9,11,11,29,53,82,129,215,389,726,1237,2079,3660,6386,11127,
%T 19719,34658,60358,105776,185641,324822,569565,999824,1753763,3075263,
%U 5390839,9452903,16579307,29065205,50947822,89330076,156628094,274559046,481250343
%N Number of gap-free Carlitz compositions of n.
%C These are integer compositions such that no adjacent parts are equal and their set of parts covers some interval.
%e a(6) = 9 counts: (1,2,1,2), (2,1,2,1), (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1), (6).
%o (PARI)
%o Ca_x(s, N)={my(x='x+O('x^N), g=if(#s <1, 1, sum(i=1, #s, (Ca_x(s[^i], N) * x^(s[i])/(1+x^(s[i]))))/(1-sum(i=1, #s, (x^(s[i]))/(1+x^(s[i])))))); return(g)}
%o B_x(N)={my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, for(k=0,N, h += Ca_x([(1+k)..(j+k)], N+1)); j++); my(a = Vec(h)); vector(N, i, a[i])}
%o B_x(20)
%Y Cf. A003242, A011782, A107428, A107429, A374147, A374727, A374728.
%K nonn
%O 1,3
%A _John Tyler Rascoe_, Jul 17 2024
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