A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

1, 1, 2, 4, 8, 17, 36, 76, 162, 345, 734, 1564, 3332, 7098, 15124, 32224, 68658, 146291, 311704, 664152, 1415124, 3015237, 6424636, 13689132, 29167776, 62148513, 132421414, 282153672, 601192008, 1280975135, 2729406380, 5815615784, 12391480916, 26402844538, 56257214530, 119868682488

Offset: 0

Author

Joerg Arndt, Jun 29 2013

Keywords

• nonn

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

Crossrefs

Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).

Cf. A129183.

Programs

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 *)

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)

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