A309536 Total number of triangular numbers in all compositions of n.

0, 1, 2, 6, 14, 33, 77, 174, 389, 860, 1885, 4098, 8853, 19020, 40668, 86593, 183698, 388421, 818892, 1721884, 3611968, 7560337, 15793474, 32932549, 68556300, 142495004, 295754816, 613039248, 1269137729, 2624393922, 5421024773, 11186523404, 23061994524

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3312

Formula

G.f.: Sum_{k>=1} x^(k*(k+1)/2)*(1-x)^2/(1-2*x)^2.

a(n) ~ c * 2^n * n, where c = 0.1604081401637884665734606925563573585565153844... - Vaclav Kotesovec, Aug 18 2019

Example

a(4) = 14: (1)(1)(1)(1), 2(1)(1), (1)2(1), (1)(1)2, 22, (3)(1), (1)(3), 4.

Maple

a:= proc(n) option remember; add(a(n-j)+
      `if`(issqr(8*j+1), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

Mathematica

CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)) - 1)*((1 - x)^2/(1 - 2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)

Crossrefs

Cf. A000217, A102291, A263235.

Sequence in context: A232497 A089351 A005380 * A257557 A124612 A230439

Adjacent sequences: A309533 A309534 A309535 * A309537 A309538 A309539

Keyword

nonn

Author

Alois P. Heinz, Aug 06 2019

Status

approved