A067726 a(n) = 6*n^2 + 12*n.

18, 48, 90, 144, 210, 288, 378, 480, 594, 720, 858, 1008, 1170, 1344, 1530, 1728, 1938, 2160, 2394, 2640, 2898, 3168, 3450, 3744, 4050, 4368, 4698, 5040, 5394, 5760, 6138, 6528, 6930, 7344, 7770, 8208, 8658, 9120, 9594, 10080, 10578, 11088, 11610

Offset: 1

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn
• easy

Comments

Positive numbers k such that 6*(6 + k) is a perfect square.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Leo Tavares, Illustration: Star Cut Hexagons
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

Cf. A003154, A003215.

Programs

GAP

List([1..45], n-> 6*n*(n+2)); # G. C. Greubel, Sep 01 2019

Magma

[6*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Maple

seq(6*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Mathematica

Select[ Range[15000], IntegerQ[ Sqrt[ 6(6 + # )]] & ]
CoefficientList[Series[6*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
6*(Range[2, 45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{18,48,90},60] (* Harvey P. Dale, May 10 2022 *)

PARI

a(n)=6*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

Sage

[6*n*(n+2) for n in (1..45)] # G. C. Greubel, Sep 01 2019

Formula

G.f.: 6*x*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Jul 08 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012

E.g.f.: 6*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/24. (End)

a(n) = A003215(2*n) - A003154(n). - Leo Tavares, May 20 2023

a(n) = 6*A005563(n). - Hugo Pfoertner, May 24 2023