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Number of compositions of n in which the least part is even.
4

%I #22 Oct 23 2024 17:01:02

%S 0,1,0,2,2,4,5,11,17,28,44,75,123,203,330,541,883,1444,2357,3848,6271,

%T 10214,16624,27051,43995,71523,116223,188790,306554,497624,807553,

%U 1310177,2125126,3446237,5587517,9057611,14680337,23789891,38546834,62449682,101163024

%N Number of compositions of n in which the least part is even.

%H Alois P. Heinz, <a href="/A103420/b103420.txt">Table of n, a(n) for n = 1..2000</a> (first 1000 terms from Robert Israel)

%F G.f.: Sum((1-x)^2*x^(2*n)/((1-x-x^(2*n))*(1-x-x^(2*n+1))), n=1..infinity).

%F G.f.: Sum(x^(2*n)/((1-x)^n*(1+x^n)),n=1..infinity). - _Vladeta Jovovic_, Mar 02 2008

%F a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-1). - _Vaclav Kotesovec_, May 01 2014

%p N:= 50: # for a(1) .. a(N)

%p G:= add(x^(2*n)/((1-x)^n*(1+x^n)),n=1..N/2):

%p S:= series(G,x,N+1):

%p [seq(coeff(S,x,i),i=1..N)]; # _Robert Israel_, Oct 23 2024

%p # second Maple program:

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

%p irem(m, 2), add(b(n-j, min(m, j)), j=1..n))

%p end:

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

%p seq(a(n), n=1..42); # _Alois P. Heinz_, Oct 23 2024

%t 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 *)

%Y Cf. A103419, A103421, A103422, A027187, A027193, A026804, A026805.

%K easy,nonn

%O 1,4

%A _Vladeta Jovovic_, Feb 04 2005

%E More terms from _Robert G. Wilson v_, Feb 05 2005