A372702 - OEIS

%I #70 May 27 2024 16:04:32

%S 6,12,32,72,152,311,625,1225,2378,4566,8700,16475,31052,58290,109079,

%T 203584,379144,704821,1308268,2425259,4491074,8308879,15360082,

%U 28376089,52391492,96683649,178344205,328854566,606190627,1117103729,2058129088,3791056189

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

%H Alois P. Heinz, <a href="/A372702/b372702.txt">Table of n, a(n) for n = 6..3779</a>

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

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

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

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

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

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

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

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

%p b:= proc(n, t) option remember; `if`(n=0, `if`(t=7, 1, 0),

%p       add(b(n-j, Bits[Or](t, 2^(j-1))), j=1..min(n, 3)))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=6..42);  # _Alois P. Heinz_, May 25 2024

%o (PARI)

%o 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)}

%o B_x(n) ={my(h=C_x([1,2,3],n)); Vec(h)}

%o B_x(40)

%Y Cf. A048793, A107429, A245738.

%Y Column k=3 of A373118.

%K nonn,easy

%O 6,1

%A _John Tyler Rascoe_, May 25 2024