A067724 a(n) = 5*n^2 + 10*n.

15, 40, 75, 120, 175, 240, 315, 400, 495, 600, 715, 840, 975, 1120, 1275, 1440, 1615, 1800, 1995, 2200, 2415, 2640, 2875, 3120, 3375, 3640, 3915, 4200, 4495, 4800, 5115, 5440, 5775, 6120, 6475, 6840, 7215, 7600, 7995, 8400, 8815, 9240, 9675

Offset: 1

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn
• easy

Comments

Positive numbers m such that 5*(5 + m) 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. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067726 (k=6), A028347 (k=4), A067725 (k=3), A054000 (k=2), A067998 (k=1).

Cf. A055998.

Programs

Magma

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

Mathematica

Select[Range[10000], IntegerQ[ Sqrt[5 (5 + # )]] &]
CoefficientList[Series[5 (3 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 08 2012 *)
Table[5n^2+10n,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{15,40,75},60] (* Harvey P. Dale, May 22 2018 *)

PARI

a(n)=5*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

Formula

From Vincenzo Librandi, Jul 08 2012: (Start)

G.f.: 5*x*(3 - x)/(1 - x)^3.

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

a(n) = A055998(3*n) + A055998(n). - Bruno Berselli, Sep 23 2016

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 3/20.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/20. (End)

E.g.f.: 5*exp(x)*x*(3 + x). - Stefano Spezia, Oct 01 2023