A142463 a(n) = 2*n^2 + 2*n - 1.

-1, 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, 4703, 4899, 5099

Offset: 0

Author

Roger L. Bagula, Sep 19 2008

Keywords

• sign
• easy

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).

Crossrefs

Cf. A000096, A005408, A132209, A144562, A188653.

Cf. A005563, A016945, A001105, A000290, A004767, A046092, A002378, A028387.

Programs

Magma

[2*n^2+2*n-1: n in [0..100]];

Maple

A142463:= n-> 2*n^2 +2*n -1; seq(A142463(n), n=0..50); # G. C. Greubel, Mar 01 2021

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 *)

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

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) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.

a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.

a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.

a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)

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

Extensions

Edited by the Associate Editors of the OEIS, Sep 02 2009