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

Author

William A. Tedeschi, Feb 29 2008

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A137928, A002061. A bisection of A137930.

Programs

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

Formula

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