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A014209 n^2 + 3*n - 1. 20
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Given the roots to n^2 + 3n - 1, a = -3.302775..., b = .302775...; then a(n) = (n + 3 + a) * (n + 3 + b). Example: a(3) = 17 = (6 - 3.302...) * (6 + .302775) -  Gary W. Adamson, Jul 29 2009

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

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

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).

FORMULA

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

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: (-1+6*x-3*x^2)/(1-x)^3. - Colin Barker, Feb 12 2012

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 *)

PROG

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

CROSSREFS

Cf. A003215, A002522.

Sequence in context: A103967 A032400 A004621 * A202253 A057258 A018466

Adjacent sequences:  A014206 A014207 A014208 * A014210 A014211 A014212

KEYWORD

sign,easy,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 22 08:03 EDT 2017. Contains 288605 sequences.