A103420 Number of compositions of n in which the least part is even.
0, 1, 0, 2, 2, 4, 5, 11, 17, 28, 44, 75, 123, 203, 330, 541, 883, 1444, 2357, 3848, 6271, 10214, 16624, 27051, 43995, 71523, 116223, 188790, 306554, 497624, 807553, 1310177, 2125126, 3446237, 5587517, 9057611, 14680337, 23789891, 38546834, 62449682, 101163024
Offset: 1
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Robert Israel)
Programs
-
Maple
N:= 50: # for a(1) .. a(N) G:= add(x^(2*n)/((1-x)^n*(1+x^n)),n=1..N/2): S:= series(G,x,N+1): [seq(coeff(S,x,i),i=1..N)]; # Robert Israel, Oct 23 2024 # second Maple program: b:= proc(n, m) option remember; `if`(n=0, 1- irem(m, 2), add(b(n-j, min(m, j)), j=1..n)) end: a:= n-> b(n, infinity): seq(a(n), n=1..42); # Alois P. Heinz, Oct 23 2024
-
Mathematica
Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n)/((1 - x - x^(2n))*(1 - x - x^(2n + 1))), {n, 40}]], {x, 0, 40}], x]] (* Robert G. Wilson v, Feb 05 2005 *)
Formula
G.f.: Sum((1-x)^2*x^(2*n)/((1-x-x^(2*n))*(1-x-x^(2*n+1))), n=1..infinity).
G.f.: Sum(x^(2*n)/((1-x)^n*(1+x^n)),n=1..infinity). - Vladeta Jovovic, Mar 02 2008
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-1). - Vaclav Kotesovec, May 01 2014
Extensions
More terms from Robert G. Wilson v, Feb 05 2005