A206739 G.f.: 1/(1 - x/(1 - x^4/(1 - x^9/(1 - x^16/(1 - x^25/(1 - x^36/(1 -...- x^(n^2)/(1 -...))))))), a continued fraction.

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 37, 52, 72, 99, 138, 193, 269, 373, 518, 722, 1006, 1399, 1944, 2705, 3766, 5241, 7290, 10141, 14112, 19638, 27323, 38012, 52889, 73593, 102398, 142470, 198225, 275809, 383760, 533954, 742923, 1033685, 1438254

Offset: 0

Author

Paul D. Hanna, Feb 12 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 +...
		

Crossrefs

Cf. A206740, A290975.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1]^2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)

Maxima

T(n, m):=if n=m then 1 else  sum(binomial(m, i)*T((n-m)/3, i), i, 1, (n-m)/3);
makelist(sum(T(n,k),k,0,n),n,0,20); /* Vladimir Kruchinin, Mar 21 2015 */

PARI

{a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^((M-k+1)^2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

PARI

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^((k+1)^2) / G(k+1) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013

Formula

a(n) = sum(k=0..n, T(n,k)), where T(n, m)=sum(i=1..(n-m)/3, binomial(m, i)*T((n-m)/3,i)), T(n,n)=1. - Vladimir Kruchinin, Mar 21 2015

G.f.: A(x)=1/B(x), where B(x) is g.f. of A290975. - Seiichi Manyama, Aug 18 2017

a(n) ~ c * d^n, where d = 1.391377080590304271048017099353... and c = 0.3625537262803710555422183139... - Vaclav Kotesovec, Aug 24 2017