A014209 a(n) = n^2 + 3*n - 1.

-1, 3, 9, 17, 27, 39, 53, 69, 87, 107, 129, 153, 179, 207, 237, 269, 303, 339, 377, 417, 459, 503, 549, 597, 647, 699, 753, 809, 867, 927, 989, 1053, 1119, 1187, 1257, 1329, 1403, 1479, 1557, 1637, 1719, 1803, 1889, 1977, 2067, 2159, 2253, 2349, 2447, 2547, 2649

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

Difference between n-th centered hexagonal number and (2*n)^2. - Alonso del Arte, Jul 06 2004

Given the roots to n^2 + 3*n - 1, a = -3.302775..., b = 0.302775...; then a(n) = (n + 3 + a)*(n + 3 + b). Example: a(3) = 17 = (6 - 3.302...)*(6 + 0.302775). - Gary W. Adamson, Jul 29 2009

a(n-1) = n*(n+1) - 3, with a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 13 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013

Numbers m >= -1 such that 4*m + 13 is a square. - Bruce J. Nicholson, Jul 17 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wikipedia, Centered Hexagonal Numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A003215, A176271.

Programs

Mathematica

Table[n^2+3*n-1, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Oct 08 2009 *)
CoefficientList[Series[(- 1 + 6 x - 3 x^2)/(1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 15 2013 *)

PARI

a(n)=n^2+3*n-1 \\ Charles R Greathouse IV, Sep 24 2015

Formula

For n > 0: a(n) = A176271(n+1,n). - Reinhard Zumkeller, Apr 13 2010

a(n) = a(n-1) + 2*n + 2, with n > 0, a(0)=-1. - Vincenzo Librandi, Nov 20 2010

From Colin Barker, Feb 12 2012: (Start)

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

G.f.: (-1+6*x-3*x^2)/(1-x)^3. (End)

Sum_{n>=0} 1/a(n) = 1/3 + tan(sqrt(13)*Pi/2)*Pi/sqrt(13). - Amiram Eldar, Jan 08 2023

E.g.f.: exp(x)*(-1 + 4*x + x^2). - Elmo R. Oliveira, Oct 31 2024