A372702 Number of compositions of n such that the set of parts is {1,2,3}.

6, 12, 32, 72, 152, 311, 625, 1225, 2378, 4566, 8700, 16475, 31052, 58290, 109079, 203584, 379144, 704821, 1308268, 2425259, 4491074, 8308879, 15360082, 28376089, 52391492, 96683649, 178344205, 328854566, 606190627, 1117103729, 2058129088, 3791056189

Offset: 6

Author

John Tyler Rascoe, May 25 2024

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..3779

Crossrefs

Cf. A048793, A107429, A245738.

Column k=3 of A373118.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, `if`(t=7, 1, 0),
      add(b(n-j, Bits[Or](t, 2^(j-1))), j=1..min(n, 3)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=6..42);  # Alois P. Heinz, May 25 2024

PARI

C_x(s,N)={my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_x(setminus(s,[s[i]]),N) * x^(s[i]) )/(1-sum(i=1,#s, x^(s[i]))))); return(g)}
B_x(n) ={my(h=C_x([1,2,3],n)); Vec(h)}
B_x(40)

Formula

G.f.: C({1,2,3},x) = (x^6/(-x^3 - x^2 - x + 1)) *

(1/((1 - x)*(-x^2 - x + 1)) +

1/((1 - x)*(-x^3 - x + 1)) +

1/((1 - x^2)*(-x^2 - x + 1)) +

1/((1 - x^2)*(-x^3 - x^2 + 1)) +

1/((1 - x^3)*(-x^3 - x + 1)) +

1/((1 - x^3)*(-x^3 - x^2 + 1))).

Where C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)).