A373420 Number of Carlitz compositions of n (see A003242) such that the first and last parts are equal.

1, 1, 1, 1, 2, 3, 2, 7, 11, 17, 26, 54, 86, 155, 272, 464, 816, 1447, 2507, 4400, 7706, 13456, 23570, 41293, 72212, 126394, 221282, 387219, 677714, 1186311, 2076170, 3633761, 6360219, 11131698, 19483066, 34100455, 59683664, 104460655, 182832044, 319999739

Offset: 0

Author

John Tyler Rascoe, Aug 16 2024

Keywords

• nonn
• easy

Examples

			a(7) = 7 counts: (1,2,1,2,1), (1,2,3,1), (1,3,2,1), (1,5,1), (2,3,2), (3,1,3), and (7).
		

Crossrefs

Cf. A001045, A003242, A285981, A291941, A374517.

Programs

PARI

C_x(N) = {my(g=1/(1-sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my( x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
D_x(N) = {my( x='x+O('x^N), f=1+sum(i=1,N, A_x(i,N))); Vec(f)}
D_x(40)

Formula

G.f.: 1 + Sum_{i>0} (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2 where C(x) is the g.f. for A003242.