A088352 G.f. = continued fraction: A(x) = 1/(1-x-x^2/(1-x^3-x^4/(1-x^5-x^6/(1-x^7-x^8/(...))))).

1, 1, 2, 3, 5, 9, 16, 28, 50, 89, 158, 282, 503, 896, 1598, 2850, 5082, 9064, 16166, 28832, 51424, 91719, 163588, 291774, 520407, 928196, 1655530, 2952805, 5266626, 9393565, 16754386, 29883166, 53299700, 95065503, 169559118, 302426167, 539408258, 962090267

Offset: 0

Author

Paul D. Hanna, Sep 26 2003

Keywords

• nonn
• easy

Comments

From Peter Bala, Jul 29 2019: (Start)

a(n) = the number of triangle stacks of large Schröder type on n triangles. See Links for a definition and an illustration.

Cf. A224704, which enumerates triangle stacks (of small Schröder type) on n triangles and A143951, which enumerates triangle stacks (of Dyck type) on n triangles. (End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300
• P. Bala, Illustration for a(5) = 9

Crossrefs

Cf. A143951, A224704.

Programs

Mathematica

nmax = 40; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^(2*k), 1 - x^(2*k + 1), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 01 2019 *)

Formula

a(n) ~ c * d^n, where d = 1.78360320457574331710673100097614660803225788206... and c = 0.4843739369092187339166963460525819972933890792971... - Vaclav Kotesovec, Jul 01 2019

From Peter Bala, Jul 29 2019: (Start)

O.g.f. as a continued fraction: A(q) = 1/(1 - q*(1 + q)/(1 - q^4/(1 - q^3*(1 + q^3)/(1 - q^8/( 1 - q^5*(1 + q^5)/(1 - q^12/( (...) ))))))).

O.g.f. as a ratio of q-series: A(q) = N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2+2*n)/( Product_{k = 1..2*n+1} (1 - q^k) ) and D(q) = Sum_{n >= 0} (-1)^n*q^(2*n^2)/( Product_{k = 1..2*n} (1 - q^k) ).

In the above asymptotic formula, 1/d = 0.5606628186... is the minimal positive real zero of D(q), and is the dominant singularity of N(q)/D(q). (End)

Extensions

More terms from Vaclav Kotesovec, Jul 01 2019