A006137 a(n) = 1 + n/2 + 9*n^2/2.

1, 6, 20, 43, 75, 116, 166, 225, 293, 370, 456, 551, 655, 768, 890, 1021, 1161, 1310, 1468, 1635, 1811, 1996, 2190, 2393, 2605, 2826, 3056, 3295, 3543, 3800, 4066, 4341, 4625, 4918, 5220, 5531, 5851, 6180, 6518, 6865, 7221, 7586, 7960, 8343, 8735, 9136, 9546

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

72*a(n) - 71 = (18*n+1)^2 = A161705(n)^2 is a perfect square. - Klaus Purath, Jan 14 2022

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen, On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force, arXiv:2106.11416 [math-ph], 2021.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002378, A003215, A161705, A276819.

Programs

Mathematica

Table[1+n/2+9 n^2/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,6,20},40] (* Harvey P. Dale, Oct 05 2012 *)

PARI

a(n)=1+n/2+9*n^2/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = a(n-1) + 9*n - 4 (with a(0)=1). - Vincenzo Librandi, Nov 18 2010

From Klaus Purath, Jan 14 2022: (Start)

a(n) = A276819(n) + n.

A003215(a(n)) - A003215(a(n)-3) = A002378(9*n). (End)

From Stefano Spezia, Dec 25 2022: (Start)

O.g.f.: (1 + 3*x+ 5*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(2 + 10*x + 9*x^2)/2. (End)