A117748 Triangular numbers divisible by 3.

0, 3, 6, 15, 21, 36, 45, 66, 78, 105, 120, 153, 171, 210, 231, 276, 300, 351, 378, 435, 465, 528, 561, 630, 666, 741, 780, 861, 903, 990, 1035, 1128, 1176, 1275, 1326, 1431, 1485, 1596, 1653, 1770, 1830, 1953, 2016, 2145, 2211, 2346, 2415, 2556, 2628, 2775

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = 3*A001318(n-1). - Michel Marcus, Apr 24 2016

From Colin Barker, Apr 24 2016: (Start)

a(n) = 3*(1-(-1)^n + 2*(-3+(-1)^n)*n + 6*n^2)/16.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.

G.f.: 3*x^2*(1+x+x^2) / ((1-x)^3*(1+x)^2).

(End)

E.g.f.: 3*(-1 - 2*x + exp(2*x) + 6*x^2*exp(2*x))*exp(-x)/16. - Ilya Gutkovskiy, Apr 24 2016

a(n) = A299412(n)/A007494(n). - Justin Gaetano, Feb 15 2018

Sum_{n>=2} 1/a(n) = 2 - 2*Pi/(3*sqrt(3)). - Amiram Eldar, Mar 24 2021

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 6, 15, 21}, 50] (* G. C. Greubel, Jun 19 2017 *)
Select[Accumulate[Range[0, 100]], Divisible[#, 3]&] (* Harvey P. Dale, Feb 11 2018 *)

Prog

(PARI) lista(nn) = {for (i = 0, nn, t = i*(i+1)/2; if (t % 3 == 0, print1(t, ", "); )); } \\ Michel Marcus, Jun 01 2013
(PARI) concat(0, Vec(3*x^2*(1+x+x^2)/((1-x)^3*(1+x)^2) + O(x^50))) \\ Colin Barker, Apr 24 2016

Crossrefs

Cf. A000217, A001318, A007494, A299412..

Sequence in context: A095869 A292610 A069559 * A061066 A093799 A087359

Adjacent sequences: A117745 A117746 A117747 * A117749 A117750 A117751

Keyword

easy,nonn

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Status

approved