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A309536 Total number of triangular numbers in all compositions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 26 12:43 EST 2020. Contains 332280 sequences. (Running on oeis4.)