A374925 Number of n-color compositions of n having at least one pair of adjacent parts that are the same color.

0, 0, 1, 3, 10, 31, 91, 259, 726, 2007, 5489, 14888, 40122, 107574, 287239, 764405, 2028679, 5371858, 14198008, 37467982, 98749767, 259984452, 683865318, 1797500121, 4721662597, 12396308875, 32531025970, 85337831350, 223794544179, 586736215856, 1537941527011

Offset: 0

Author

John Tyler Rascoe, Jul 24 2024

Keywords

• nonn
• easy

Examples

			a(4) = 10 counts: (1,1,1,1), (1,1,2_a), (1,1,2_b), (1,2_a,1), (1,3_a), (2_a,1,1), (2_a,2_a), (2_b,1,1), (2_b,2_b), (3_a,1).
		

Crossrefs

Cf. A003242, A088305, A242551, A330774, A374727, A374728.

Programs

PARI

C_x(N) = {my(x='x+O('x^N), h=(sum(i=1,N,(x^(2*i))/((1-x)*(1-x+x^i)*(1-sum(j=1,N, (x^j)/(1-x+x^j))))))/(1-sum(i=1,N,(x^i)/(1-x)))); concat([0,0],Vec(h))}
C_x(40)

Formula

G.f.: Sum_{i>0} ( x^(2*i)/((1 - x)*(1 - x + x^i)*(1 - Sum_{j>0} ((x^j)/(1 - x + x^j)))) )/( 1 - Sum_{k>0} ((x^k)/(1 - x)) ).

a(n) = A088305(n) - A242551(n).