A085046 a(n) = n^2 - (1 + (-1)^n)/2.

1, 3, 9, 15, 25, 35, 49, 63, 81, 99, 121, 143, 169, 195, 225, 255, 289, 323, 361, 399, 441, 483, 529, 575, 625, 675, 729, 783, 841, 899, 961, 1023, 1089, 1155, 1225, 1295, 1369, 1443, 1521, 1599, 1681, 1763, 1849, 1935, 2025, 2115, 2209, 2303, 2401, 2499, 2601

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Keywords

• nonn
• easy

Comments

Sequence pattern looks like this 1*1, 1*3, 3*3, 3*5, 5*5, 5*7, 7*7, 7*9, 9*9, 9*11, 11*11, ... = A109613(n-1)*A109613(n).

a(n+1) is the determinant of the n X n matrix M_(i, i)=3, M_(i, j)=2 if (i+j) is even, M_(i, j)=0 if (i+j) is odd. - Benoit Cloitre, Aug 06 2003

a(n) is also the longest path, in number of cells, between diagonally opposite corners of an n X n matrix if diagonal movement between adjacent cells is not allowed and no cell is used more than once. - Ray G. Opao, Jul 02 2007

(-1)^n*a(n) appears to be the Hankel transform of A141222. - Paul Barry, Jun 14 2008

Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. 4*a(n) is the total number of unit edges in each figure (see example and cf. A255840, A255876). - Wesley Ivan Hurt, Mar 09 2015

Examples

			4*a(n) is the number of unit edges in the pattern below (see comments).
                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5
- _Wesley Ivan Hurt_, Mar 09 2015
		

Links

• Stefano Spezia, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A109613. [Bruno Berselli, Sep 17 2013]

Cf. A008811, A141222, A255840, A255876.

Programs

Magma

[n^2-(1+(-1)^n)/2 : n in [1..100]]; // Wesley Ivan Hurt, Mar 09 2015

Maple

A085046:=n->n^2-(1+(-1)^n)/2: seq(A085046(n), n=1..100); # Wesley Ivan Hurt, Mar 09 2015

Mathematica

Table[n^2-1/2 (1+(-1)^n), {n, 60}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{2,0,-2,1},{1,3,9,15},70] (* Harvey P. Dale, Oct 25 2015 *)

Formula

a(1) = 1, a(2) = 3, then a(2n) = (a(2n-1)*a(2n+1))^1/2 and a(2n+1) = {a(2n) + a(2n+2)}/2. Even-indexed terms are the geometric mean, and odd-indexed terms are the arithmetic mean, of their neighbors.

a(2n+1) = (2n+1)^2 and a(2n) = 4n^2 - 1.

a(n) = A008811(2n) - 1. - N. J. A. Sloane, Jun 12 2004

From Bruno Berselli, Sep 17 2013: (Start)

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

a(n) = n^2 - (1 + (-1)^n)/2. (End)

a(1)=1, a(2)=3, a(3)=9, a(4)=15, a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Oct 25 2015

E.g.f.: 1 - cosh(x) + x*(1 + x)*(cosh(x) + sinh(x)). - Stefano Spezia, May 26 2021

Sum_{n>=1} 1/a(n) = Pi^2/8 + 1/2. - Amiram Eldar, Aug 25 2022

Extensions

More terms from Benoit Cloitre, Aug 06 2003

Formula added in the first comment by Bruno Berselli, Sep 17 2013

Replaced name with Sep 17 2013 formula from Bruno Berselli [Wesley Ivan Hurt, May 17 2020]