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

0,3

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

Links

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

Formula

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

Example

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.

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]

Prog

(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

Crossrefs

Cf. A088305, A351638.

Sequence in context: A190512 A110513 A018115 * A018007 A295091 A071879

Adjacent sequences: A242548 A242549 A242550 * A242552 A242553 A242554

Keyword

nonn,easy

Author

Geoffrey Critzer, May 17 2014

Status

approved