A140673 a(n) = 3*n*(n + 5)/2.

0, 9, 21, 36, 54, 75, 99, 126, 156, 189, 225, 264, 306, 351, 399, 450, 504, 561, 621, 684, 750, 819, 891, 966, 1044, 1125, 1209, 1296, 1386, 1479, 1575, 1674, 1776, 1881, 1989, 2100, 2214, 2331, 2451, 2574, 2700, 2829, 2961, 3096

Offset: 0

Author

Omar E. Pol, May 22 2008

Keywords

• nonn
• easy

Comments

a(n) equals the number of vertices of the A256666(n)-th graph (see Illustration of initial terms in A256666 Links). - Ivan N. Ianakiev, Apr 20 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A055998.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Programs

Mathematica

Table[Sum[i + n - 3, {i, 6, n}], {n, 5, 52}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[3 n (n + 5)/2, {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3,-3,1},{0,9,21},50] (* Harvey P. Dale, Jul 20 2023 *)

PARI

concat(0, Vec(3*x*(3 - 2*x)/(1 - x)^3 + O(x^100))) \\ Michel Marcus, Apr 20 2015

PARI

a(n) = 3*n*(n+5)/2; \\ Altug Alkan, Sep 05 2018

Formula

a(n) = A055998(n)*3 = (3*n^2 + 15*n)/2 = n*(3*n + 15)/2.

a(n) = 3*n + a(n-1) + 6 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

G.f.: 3*x*(3 - 2*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011

E.g.f.: (1/2)*(3*x^2 + 18*x)*exp(x). - G. C. Greubel, Jul 17 2017

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 137/450.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 47/450. (End)