A067725 a(n) = 3*n^2 + 6*n.

0, 9, 24, 45, 72, 105, 144, 189, 240, 297, 360, 429, 504, 585, 672, 765, 864, 969, 1080, 1197, 1320, 1449, 1584, 1725, 1872, 2025, 2184, 2349, 2520, 2697, 2880, 3069, 3264, 3465, 3672, 3885, 4104, 4329, 4560, 4797, 5040, 5289, 5544, 5805, 6072, 6345, 6624

Offset: 0

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn
• easy

Comments

Numbers h such that 3*(3 + h) is a perfect square. - Alexander D. Healy, Tj Tullo, Avery Pickford, Sep 20 2004

Equivalently, numbers k such that k/3+1 is a square. - Bruno Berselli, Apr 10 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005563.

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067726 (k=6), A067724 (k=5), A028347 (k=4), A054000 (k=2), A005563 (k=1).

Programs

GAP

List([0..50], n-> 3*n*(n+2)); # G. C. Greubel, Sep 01 2019

Magma

[3*n*(n+2): n in [0..50]]; // G. C. Greubel, Sep 01 2019

Maple

seq(3*n*(n+2), n=0..50); # G. C. Greubel, Sep 01 2019

Mathematica

Select[ Range[10000], IntegerQ[ Sqrt[ 3(3 + # )]] & ]
3*(Range[50]^2 -1) (* G. C. Greubel, Sep 01 2019 *)

PARI

a(n)=3*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

Sage

[3*n*(n+2) for n in (0..50)] # G. C. Greubel, Sep 01 2019

Formula

a(n) = 3*A005563(n). - Zerinvary Lajos, Mar 06 2007

a(n) = a(n-1) + 6*n + 3, with n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010

From Colin Barker, Apr 11 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: 3*x*(3-x)/(1-x)^3. (End)

E.g.f.: 3*x*(x + 3)*exp(x). - G. C. Greubel, Jul 20 2017

From Amiram Eldar, Feb 26 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/12. (End)

Extensions

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar