A085789 Partial sums of n 3-spaced triangular numbers beginning with t(2), e.g., a(2) = t(2) + t(5) = 3 + 15 = 18.

3, 18, 54, 120, 225, 378, 588, 864, 1215, 1650, 2178, 2808, 3549, 4410, 5400, 6528, 7803, 9234, 10830, 12600, 14553, 16698, 19044, 21600, 24375, 27378, 30618, 34104, 37845, 41850, 46128, 50688, 55539, 60690, 66150, 71928, 78033, 84474, 91260, 98400, 105903

Offset: 1

Author

Jon Perry, Jul 23 2003

Keywords

• nonn
• easy

Comments

Sums of rows of triangle A100345 (n>0).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A002411, A004188, A100345.

Programs

Magma

[3/2*n^2*(n+1): n in [1..40]]; // Vincenzo Librandi, Aug 14 2017

Mathematica

CoefficientList[Series[3 (1 + 2 x) / (1 - x)^4, {x, 0, 40}], x](* Vincenzo Librandi, Aug 14 2017 *)
LinearRecurrence[{4,-6,4,-1},{3,18,54,120},50] (* Harvey P. Dale, May 14 2023 *)

Formula

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

a(n) = 3*n*binomial(n+1,2) = 3*n*A000217(n) = 3*A002411(n). - Arkadiusz Wesolowski, Feb 10 2012

G.f.: 3*(x + 2*x^2)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012

From Amiram Eldar, Jun 29 2025: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/9 - 2/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/18 - 4*log(2)/3 + 2/3. (End)

Extensions

More terms from Reinhard Zumkeller, Nov 18 2004