A140672 a(n) = n*(3*n + 13)/2.

0, 8, 19, 33, 50, 70, 93, 119, 148, 180, 215, 253, 294, 338, 385, 435, 488, 544, 603, 665, 730, 798, 869, 943, 1020, 1100, 1183, 1269, 1358, 1450, 1545, 1643, 1744, 1848, 1955, 2065, 2178, 2294, 2413, 2535, 2660, 2788, 2919, 3053

Offset: 0

Author

Omar E. Pol, May 22 2008

Keywords

• easy
• nonn

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

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

Magma

[(3*n^2 + 13*n)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 27 2023

Mathematica

Table[n (3 n + 13)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 8, 19}, 50] (* Harvey P. Dale, Dec 16 2011 *)

PARI

a(n)=n*(3*n+13)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = (3*n^2 + 13*n)/2.

a(n) = 3*n + a(n-1) + 5 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

a(0)=0, a(1)=8, a(2)=19; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 16 2011

G.f.: x*(8 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011

E.g.f.: (1/2)*(3*x^2 +16*x)*exp(x). - G. C. Greubel, Jul 17 2017