A374728 Number of n-color gap-free compositions of n.

1, 1, 1, 3, 7, 19, 45, 105, 239, 507, 1079, 2303, 4829, 10425, 23263, 53363, 127995, 318983, 816057, 2133241, 5640135, 14975051, 39772751, 105322879, 277547989, 727276225, 1894282195, 4903985955, 12621154315, 32302574959, 82248961437, 208426306113, 525884062427

Offset

1,4

Comments

These are integer compositions whose set of parts covers some interval and contains k colors of each part k.

Links

Table of n, a(n) for n=1..33.

Example

a(5) = 7 counts: (1,1,1,1,1), (1,2_a,2_b), (1,2_b,2_a), (2_a,1,2_b), (2_a,2_b,1), (2_b,1,2_a), (2_b,2_a,1).

Prog

(PARI)
colr(x, y)={my(r=y-x+1, v=[x..y], z = vector(r*(r+(1+(x-1)*2))/2), k=1); for(i=1, #v, for(j=1, v[i], z[k]=v[i]; k++)); return(z)}
C_x(s, N)={my(x='x+O('x^N), g=if(#s <1, 1, sum(i=1, #s, C_x(s[^i], N+1) * x^(s[i]) )/(1-sum(i=1, #s, x^(s[i]))))); return(g)}
B_x(N)={my(x='x+O('x^N), h=0); for(u=1, N, my(j=0); while(vecsum(colr(u, u+j)) <= N, h += C_x(colr(u, u+j), N+1); j++)); my(a = Vec(h)); vector(N, i, a[i])}
B_x(20)

Crossrefs

Cf. A003242, A011782, A107428, A107429, A374147, A374726, A374727.

Sequence in context: A146653 A096447 A274596 * A348008 A185696 A141344

Adjacent sequences: A374725 A374726 A374727 * A374729 A374730 A374731

Keyword

nonn

Author

John Tyler Rascoe, Jul 17 2024

Status

approved