A114254 Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral.

1, 25, 101, 261, 537, 961, 1565, 2381, 3441, 4777, 6421, 8405, 10761, 13521, 16717, 20381, 24545, 29241, 34501, 40357, 46841, 53985, 61821, 70381, 79697, 89801, 100725, 112501, 125161, 138737, 153261, 168765, 185281, 202841, 221477, 241221

Offset: 0

Author

William A. Tedeschi, Feb 06 2008, Mar 01 2008

Keywords

• easy
• nonn

Examples

			For n = 1, the 3 X 3 spiral is
.
       7---8---9
       |
       6   1---2
       |       |
       5---4---3
.
so a(1) = 7 + 9 + 1 + 5 + 3 = 25.
.
For n = 2, the 5 X 5 spiral is
.
  21--22--23--24--25
   |
  20   7---8---9--10
   |   |           |
  19   6   1---2  11
   |   |       |   |
  18   5---4---3  12
   |               |
  17--16--15--14--13
.
so a(2) = 21 + 25 + 7 + 9 + 1 + 5 + 3 + 17 + 13 = 101.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Project Euler, Problem 28. Number Spiral Diagonals
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A016754, A054569, A053755, A054554 for diagonals from origin.

Cf. A325958 (first differences).

Programs

Mathematica

Array[1 + 10 #^2 + (16 #^3 + 26 #)/3 &, 36, 0] (* Michael De Vlieger, Mar 01 2018 *)

PARI

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3; \\ Joerg Arndt, Mar 01 2018

Formula

O.g.f.: 3/(-1+x) + 16/(-1+x)^2 + 44/(-1+x)^3 + 32/(-1+x)^4 = (1 + 21*x + 7*x^2 + 3*x^3)/(-1+x)^4. - R. J. Mathar, Feb 10 2008

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3. [Corrected by Arie Groeneveld, Aug 17 2008]