A059924 Write the numbers from 1 to n^2 in a spiraling square; a(n) is the total of the sums of the two diagonals.

0, 2, 10, 34, 80, 158, 274, 438, 656, 938, 1290, 1722, 2240, 2854, 3570, 4398, 5344, 6418, 7626, 8978, 10480, 12142, 13970, 15974, 18160, 20538, 23114, 25898, 28896, 32118, 35570, 39262, 43200, 47394, 51850, 56578, 61584, 66878, 72466, 78358, 84560, 91082, 97930

Offset

0,2

Comments

If n is odd, n^2 is counted twice.

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = 3a(n-1)-2a(n-2)-2a(n-3)+3a(n-4)-a(n-5), a(0) = 0, a(1) = 2, a(2) = 10, a(3) = 34, a(4) = 80.

a(n) = (16*n^3 - 6*n^2 + 8*n + 3 - 3*(-1)^n)/12. - Frank Ellermann, Mar 16 2002

O.g.f.: (2*x+4*x^2+8*x^3+2*x^4)/(1-3*x+2*x^2+2*x^3-3*x^4+x^5)=(2*x+4*x^2+8*x^3+2*x^4)/((1-x)^4*(1+x)). - Eric Werley, Jun 30 2011

Example

Write the numbers from 1 to 16 like this:
.
   1---2---3---4
               |
  12--13--14   5
   |       |   |
  11  16--15   6
   |           |
  10---9---8---7
.
The two diagonals add to 36 and 44, so a(4) = 36 + 44 = 80.

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 2, 10, 34, 80}, 40] (* Harvey P. Dale, Mar 23 2012 *)

Prog

(PARI) a(n) = { (16*n^3 - 6*n^2 + 8*n + 3 - 3*(-1)^n)/12 } \\ Harry J. Smith, Jun 30 2009

Crossrefs

Sequence in context: A043004 A316172 A108100 * A304159 A211905 A022498

Adjacent sequences: A059921 A059922 A059923 * A059925 A059926 A059927

Keyword

easy,nice,nonn,changed

Author

Fabian Rothelius, Feb 10 2001

Extensions

Corrected and extended by Eric Werley, Jun 30 2011

Status

approved