A374726 Number of gap-free Carlitz compositions of n.

1, 1, 3, 2, 4, 9, 11, 11, 29, 53, 82, 129, 215, 389, 726, 1237, 2079, 3660, 6386, 11127, 19719, 34658, 60358, 105776, 185641, 324822, 569565, 999824, 1753763, 3075263, 5390839, 9452903, 16579307, 29065205, 50947822, 89330076, 156628094, 274559046, 481250343

Offset

1,3

Comments

These are integer compositions such that no adjacent parts are equal and their set of parts covers some interval.

Links

Table of n, a(n) for n=1..39.

Example

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).

Prog

(PARI)
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)}
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])}
B_x(20)

Crossrefs

Cf. A003242, A011782, A107428, A107429, A374147, A374727, A374728.

Sequence in context: A338246 A368225 A371267 * A084793 A033820 A095259

Adjacent sequences: A374723 A374724 A374725 * A374727 A374728 A374729

Keyword

nonn

Author

John Tyler Rascoe, Jul 17 2024

Status

approved