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

0,2

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).

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]

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A325958 (first differences).

Sequence in context: A356533 A221274 A042220 * A042222 A158551 A044276

Adjacent sequences: A114251 A114252 A114253 * A114255 A114256 A114257

Keyword

easy,nonn

Author

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

Status

approved