A067727 a(n) = 7*n^2 + 14*n.

21, 56, 105, 168, 245, 336, 441, 560, 693, 840, 1001, 1176, 1365, 1568, 1785, 2016, 2261, 2520, 2793, 3080, 3381, 3696, 4025, 4368, 4725, 5096, 5481, 5880, 6293, 6720, 7161, 7616, 8085, 8568, 9065, 9576, 10101, 10640, 11193, 11760, 12341, 12936

Offset: 1

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn
• easy

Comments

Positive numbers k such that 7*(7 + k) is a perfect square.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A186029.

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067726 (k=6), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

Programs

GAP

List([1..45], n-> 7*n*(n+2)); # G. C. Greubel, Sep 01 2019

Magma

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

Maple

seq(7*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Mathematica

Select[ Range[15000], IntegerQ[ Sqrt[ 7(7 + # )]] & ]
CoefficientList[Series[7*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
7*(Range[2,45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{21,56,105},50] (* Harvey P. Dale, Dec 07 2022 *)

PARI

a(n)= 7*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

Sage

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

Formula

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

G.f.: 7*x*(3-x)/(1-x)^3. - Vincenzo Librandi, Jul 08 2012

E.g.f.: 7*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 3/28.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/28. (End)

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010