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

0,2

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).

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)

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 *)

Prog

(PARI) a(n)=3*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011
(Magma) [3*n*(n+2): n in [0..50]]; // G. C. Greubel, Sep 01 2019
(Sage) [3*n*(n+2) for n in (0..50)] # G. C. Greubel, Sep 01 2019
(GAP) List([0..50], n-> 3*n*(n+2)); # G. C. Greubel, Sep 01 2019

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).

Sequence in context: A262044 A362420 A097658 * A213903 A351043 A001106

Adjacent sequences: A067722 A067723 A067724 * A067726 A067727 A067728

Keyword

nonn,easy

Author

Robert G. Wilson v, Feb 05 2002

Extensions

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

Status

approved