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

Author

John Tyler Rascoe, Jul 17 2024

Keywords

• nonn

Comments

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

Examples

			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).
		

Crossrefs

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

Programs

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)