A137931 Sum of the principal diagonals of a 2n X 2n square spiral.

0, 10, 56, 170, 384, 730, 1240, 1946, 2880, 4074, 5560, 7370, 9536, 12090, 15064, 18490, 22400, 26826, 31800, 37354, 43520, 50330, 57816, 66010, 74944, 84650, 95160, 106506, 118720, 131834, 145880, 160890, 176896, 193930, 212024, 231210, 251520, 272986, 295640

Offset

0,2

Comments

This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = -1 + n + Sum_{k=0..2*n} (2*k^2 - k + 1) = n - 1 + (2*n+1)*(8*n^2-n+3)/3.

a(n) = 2*n^2 + 2*n + (16*n^3 + 2*n)/3 = 2*n*(8*n^2+3*n+4)/3.

G.f.: 2*x*(3*x+5)*(x+1)/(x-1)^4. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From Elmo R. Oliveira, May 26 2026: (Start)

E.g.f.: 2*exp(x)*x*(15 + 27*x + 8*x^2)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Example

Example with n = 2:
.
   7---8---9--10
   |           |
   6   1---2  11
   |       |   |
   5---4---3  12
               |
  16--15--14--13
.
a(0) = 2(0)^2 + 2(0) + (16(0)^3 + 2(0))/3 = 0.
a(2) = 2(2)^2 + 2(2) + (16(2)^3 + 2(2))/3 = 56.

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {0, 10, 56, 170}, 50] (* Paolo Xausa, May 16 2026 *)

Prog

(Python) f = lambda n: -1 + n + sum(2*k**2 - k + 1 for k in range(0, 2*n+1))
(Python) a = lambda n: 2*n**2 + 2*n + (16*n**3 + 2*n)/3

Crossrefs

A bisection of A137930.

Cf. A002061, A137928.

Sequence in context: A202071 A281207 A228888 * A053493 A198833 A268462

Adjacent sequences: A137928 A137929 A137930 * A137932 A137933 A137934

Keyword

nonn,easy,changed

Author

William A. Tedeschi, Feb 29 2008

Status

approved