A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).

Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).

a(n) ~ c * d^sqrt(n) / n^(3/4), where d = 7.1578741786143524880205016499891016... and c = 0.10511708841962944170826735560432...

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Crossrefs

Cf. A001935, A003106, A053261, A066447, A003114, A306734, A333179.

Sequence in context: A056697 A132427 A176975 * A098523 A350514 A308620

Adjacent sequences: A333371 A333372 A333373 * A333375 A333376 A333377

Keyword

nonn

Author

Vaclav Kotesovec, Mar 17 2020

Status

approved