A206740 G.f.: 1/(1 - x/(1 - x^3/(1 - x^6/(1 - x^10/(1 - x^15/(1 - x^21/(1 -...- x^(n*(n+1)/2)/(1 -...))))))), a continued fraction.

1, 1, 1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 44, 66, 99, 147, 220, 329, 490, 732, 1095, 1634, 2440, 3646, 5444, 8130, 12146, 18139, 27089, 40463, 60434, 90258, 134811, 201349, 300721, 449153, 670844, 1001939, 1496467, 2235080, 3338227, 4985868, 7446739, 11122179

Offset

0,5

Links

Table of n, a(n) for n=0..43.

Formula

G.f.: 1/Q(0) , where Q(k) = 1 - x^((2*k+1)*(2*k+2)/2)/(1 - x^((2*k+2)*(2*k+3)/2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 10 2013

a(n) ~ c * d^n, where d = 1.49356638691558702616975760297981328... and c = 0.35853801643147450974166770910994348... - Vaclav Kotesovec, Aug 24 2017

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 9*x^8 +...

Mathematica

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

Prog

(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)*(M-k+2)/2)*CF)); polcoeff(CF, n, x)}
for(n=0, 55, print1(a(n), ", "))

Crossrefs

Cf. A206739.

Sequence in context: A136423 A215245 A078932 * A172161 A117791 A022860

Adjacent sequences: A206737 A206738 A206739 * A206741 A206742 A206743

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved