A099823 G.f. is the continued fraction: A(x) = 1/[1 - x/[1 - (x-x^2)/[1 - (x^2-x^4)/[1 - (x^3-x^6)/[1-... - (x^n-x^(2n))/[1 - ... ]]]]]]].

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 67, 101, 153, 230, 346, 520, 780, 1171, 1755, 2631, 3942, 5905, 8846, 13247, 19839, 29707, 44482, 66604, 99722, 149309, 223546, 334692, 501096, 750226, 1123216, 1681635, 2517676, 3769356, 5643307, 8448900, 12649289

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Formula

G.f.: A(x) = 1/(1-x*Sum_{n=0..inf} (-1)^n*[x^((3n+2)n) + x^((3n+1)(n+1))] ).

a(n) = Sum_{k=0..m} (-1)^[n/2] * a(n-1 - A001082(k)), where A001082(m) < n <= A001082(m+1).

a(n) ~ c * d^n, where d = 1.49715009102318386309178199346058484192024290343171764086663161717870056467... and c = 1.23439530191230647588567129689364633472692295156374785190689889414622... - Vaclav Kotesovec, Jul 01 2019

Example

A(x) = 1/(1 - x*(1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 ++-- ... + (-1)^[n/2]*x^A001082(n) +...)).
a(n) = a(n-1) + a(n-2) - a(n-6) - a(n-9) + a(n-17) + a(n-22) --++...

Mathematica

nmax = 50; CoefficientList[Series[1/(1 - x*Sum[(-1)^k*(x^((3*k+2)*k) + x^((3*k+1)*(k+1))), {k, 0, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 01 2019 *)

Crossrefs

Cf. A001082.

Sequence in context: A327421 A124062 A274199 * A358335 A240523 A023436

Adjacent sequences: A099820 A099821 A099822 * A099824 A099825 A099826

Keyword

nonn

Author

Paul D. Hanna, Oct 26 2004

Status

approved