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

0,4

Links

Table of n, a(n) for n=0..30.

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).

Example

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).

Prog

(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)

Crossrefs

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

Sequence in context: A290061 A212031 A339032 * A033121 A180432 A237930

Adjacent sequences: A374919 A374920 A374924 * A374926 A374927 A374928

Keyword

nonn,easy

Author

John Tyler Rascoe, Jul 24 2024

Status

approved