A140090 a(n) = n*(3*n + 7)/2.

0, 5, 13, 24, 38, 55, 75, 98, 124, 153, 185, 220, 258, 299, 343, 390, 440, 493, 549, 608, 670, 735, 803, 874, 948, 1025, 1105, 1188, 1274, 1363, 1455, 1550, 1648, 1749, 1853, 1960, 2070, 2183, 2299, 2418, 2540, 2665, 2793, 2924

Offset: 0

Author

Omar E. Pol, May 22 2008

Keywords

• easy
• nonn

Comments

This sequence is mentioned in the Guo-Niu Han's paper, chapter 6: Dictionary of the standard puzzle sequences, p. 19 (see link). - Omar E. Pol, Oct 28 2011

Number of cards needed to build an n-tier house of cards with a flat, one-card-wide roof. - Tyler Busby, Dec 28 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
• Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains, arXiv:2506.10856 [math.PR], 2025. See p. 33.
• 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, this sequence, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Cf. numbers of the form n*(d*n + 10 - d)/2: A008587, A056000, A028347, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.

Programs

Mathematica

LinearRecurrence[{3,-3,1},{0,5,13},50] (* Harvey P. Dale, Jan 17 2022 *)

PARI

a(n)=n*(3*n+7)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: x*(5 - 2*x)/(1 - x)^3. - Bruno Berselli, Feb 11 2011

a(n) = (3*n^2 + 7*n)/2.

a(n) = a(n-1) + 3*n + 2 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010

E.g.f.: (1/2)*(3*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = 117/98 - Pi/(7*sqrt(3)) - 3*log(3)/7.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(7*sqrt(3)) + 4*log(2)/7 - 75/98. (End)