OFFSET
0,3
COMMENTS
What does this sequence count?
Conjectures from John Tyler Rascoe, Nov 04 2023: (Start)
a(n) is the number of integer compositions of n into two kinds of odd parts with the following restrictions. Each composition has first part 1a. For all a parts pa_i >= px_{i+1} and for all b parts pb_i >= px_{i+1} or pb_i = (p+2)a_{i+1}.
In general if B(i) = b_1, b_2, ..., b_i is an infinite sequence where b_1 > 0 and b_i <= b_{i+1} for all i, let A(q) = 1/(1-q^b_1/(1-q^b_2/(1-q^b_3/(1-...)))) be a generating function where the exponents of q are the sequence B(i).
Then A(q) counts integer compositions into parts b_i with the following restrictions. Every composition has first part p_1 = b_1 and for every pair of parts (p_j,p_{j+1}), B^-1(p_j) + 1 >= B^-1(p_{j+1}). Where j is the position of the part p_j within the composition itself and B^-1(p_j) is the index of p_j in B(i). (End)
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/(1-q/(1-q/(1-q^3/(1-q^3/(1-q^5/(1-q^5/(1-q^7/(1-q^7/(1-...))))))))).
G.f.: 1/W(0), where W(k)= 1 - x^(2*k+1)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - Vaclav Kotesovec, Sep 05 2017
Conjecture: a(n) = Sum_{i=0..floor((n-sqrt(2*n-1))/2)} A129183(n-(2*i),n-i). - John Tyler Rascoe, Nov 04 2023
MATHEMATICA
nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(2*Range[nmax + 1] - 2*Floor[Range[nmax + 1]/2] - 1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)
PROG
(PARI) N = 66; q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
gf = 1 / G(0)
Vec(gf)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 29 2013
STATUS
approved