A132209 a(0) = 0 and a(n) = 2*n^2 + 2*n - 1, for n>=1.

0, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511

Offset

0,2

Comments

Previous name was: Sequence gives X values that satisfy the integer equation 2*X^3 + 3*X^2 = Y^2.

To find Y values: b(n) = (2*n^2 + 2*n - 1)*(2*n - 1).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Formula

a(n) = 2*n^2 + 2*n - 1 for n>=1.

G.f.: x*(1+x)*(3-x)/(1-x)^3. - R. J. Mathar, Nov 14 2007

E.g.f.: 1 + (2*x^2 + 4*x -1)*exp(x). - G. C. Greubel, Jul 13 2017

From Amiram Eldar, Mar 07 2021: (Start)

Sum_{n>=1} 1/a(n) = 1 + sqrt(3)*Pi*tan(sqrt(3)*Pi/2)/6.

Product_{n>=1} (1 + 1/a(n)) = -Pi*sec(sqrt(3)*Pi/2)/2.

Product_{n>=1} (1 - 1/a(n)) = cos(sqrt(5)*Pi/2)*sec(sqrt(3)*Pi/2)/2. (End)

Mathematica

Join[{0}, LinearRecurrence[{3, -3, 1}, {3, 11, 23}, 40]] (* Vincenzo Librandi, Sep 22 2015 *)

Prog

(Magma) [0] cat [2*n^2+2*n-1: n in [1..50]]; // Vincenzo Librandi, Sep 22 2015
(PARI) for(n=0, 50, print1(if(n==0, 0, 2*n^2 + 2*n -1), ", ")) \\ G. C. Greubel, Jul 13 2017

Crossrefs

Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475, A000217.

Sequence in context: A320901 A119173 A106201 * A142463 A289575 A086497

Adjacent sequences: A132206 A132207 A132208 * A132210 A132211 A132212

Keyword

nonn

Author

Mohamed Bouhamida, Nov 06 2007

Extensions

Edited by the Associate Editors of the OEIS, Nov 15 2009

More terms from Vincenzo Librandi, Sep 22 2015

Shorter name (using formula given) from Joerg Arndt, Sep 27 2015

Status

approved