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

1,2

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

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).

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A008811, A141222, A255840, A255876.

Sequence in context: A357167 A099989 A209980 * A138495 A055927 A316261

Adjacent sequences: A085043 A085044 A085045 * A085047 A085048 A085049

Keyword

nonn,easy

Author

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

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]

Status

approved