OFFSET
0,2
COMMENTS
Essentially the same as A132209.
From Vincenzo Librandi, Nov 25 2010: (Start)
Numbers k such that 2*k + 3 is a square.
First diagonal of A144562. (End)
The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - Klaus Purath, Aug 31 2023
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Leo Tavares, Illustration: Hexagonic Diamonds.
Leo Tavares, Illustration: Hexagonic Rectangles.
Leo Tavares, Illustration: Hexagonic Crosses.
Leo Tavares, Illustration: Hexagonic Columns.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = a(n-1) + 4*n.
From Paul Barry, Nov 03 2009: (Start)
G.f.: (1 - 6*x + x^2)/(1-x)^3.
a(n) = 4*C(n+1,2) - 1. (End)
a(n) = -A188653(2*n+1). - Reinhard Zumkeller, Apr 13 2011
a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - Gary Detlefs, Oct 18 2011
a(n) = (A005408(n)^2 - 3)/2. - Zhandos Mambetaliyev, Feb 11 2017
E.g.f.: (-1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Mar 01 2021
From Leo Tavares, Nov 22 2021: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021
Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022
MAPLE
MATHEMATICA
Array[ -#*(2-#*2)-1&, 5!, 1] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
Table[2n^2+2n-1, {n, 0, 50}] (* Harvey P. Dale, Feb 29 2024 *)
PROG
(Magma) [2*n^2+2*n-1: n in [0..100]]
(PARI) a(n)=2*n^2+2*n-1 \\ Charles R Greathouse IV, Sep 24 2015
(Sage) [2*n^2 +2*n -1 for n in (0..50)] # G. C. Greubel, Mar 01 2021
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roger L. Bagula, Sep 19 2008
EXTENSIONS
Edited by the Associate Editors of the OEIS, Sep 02 2009
STATUS
approved