A242551 Number of n-length words on infinite alphabet {1,2,...} such that the maximal runs of consecutive equal integers have lengths that are at least as great as the integer.

1, 1, 2, 5, 11, 24, 53, 118, 261, 577, 1276, 2823, 6246, 13819, 30572, 67635, 149630, 331029, 732344, 1620187, 3584388, 7929844, 17543415, 38811782, 85864379, 189960150, 420254129, 929739922, 2056889538, 4550514023, 10067228909, 22272010878, 49272989918, 109008007822, 241161451563, 533528195645

Offset: 0

Author

Geoffrey Critzer, May 17 2014

Keywords

• nonn
• easy

Comments

In other words, there is no restriction on the length of runs of 1's, the length of runs of 2's must be at least two, the length of runs of 3's must be at least three...

a(n) is the number of n-color integer compositions of n such that no adjacent parts are the same color. - John Tyler Rascoe, Jul 23 2024

Examples

			a(3)=5 because we have: 111, 122, 221, 222, 333.
a(4)=11 because we have:  1111, 1122, 1221, 1222, 2211, 2221, 2222, 3331, 1333, 3333, 4444.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A088305, A351638.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1,
      `if`(t=0, 0, b(n-1, t)) +add(
      `if`(t=j, 0, b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 07 2015

Mathematica

n=nn=35;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->z^i/(1-z),{i,1,n}]),{z,0,nn}],z]

PARI

C_x(N)={my(x='x+O('x^N), h = 1/(1-sum(i=1,N, x^i/(1 - x + x^i)))); Vec(h)}
C_x(40) \\ John Tyler Rascoe, Jul 23 2024

Formula

G.f.: 1/(1 - Sum_{i>0} x^i/(1 - x + x^i)). - John Tyler Rascoe, Jul 23 2024