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

Author

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

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000217, A001318, A007494, A299412..

Programs

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 *)

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

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