A137928 The even principal diagonal of a 2n X 2n square spiral.

2, 4, 10, 16, 26, 36, 50, 64, 82, 100, 122, 144, 170, 196, 226, 256, 290, 324, 362, 400, 442, 484, 530, 576, 626, 676, 730, 784, 842, 900, 962, 1024, 1090, 1156, 1226, 1296, 1370, 1444, 1522, 1600, 1682, 1764, 1850, 1936, 2026, 2116, 2210, 2304, 2402, 2500, 2602, 2704, 2810

Offset: 1

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(1) = 2(1) + 4*floor((1-1)/4) = 2;
a(2) = 2(2) + 4*floor((2-1)/4) = 4.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A000982, A002061 (odd diagonal), A002620, A080335, A171218.

Programs

Maple

A137928:=n->2*ceil(n^2/2): seq(A137928(n), n=1..100); # Wesley Ivan Hurt, Jul 25 2017

Mathematica

LinearRecurrence[{2,0,-2,1},{2,4,10,16},60] (* Harvey P. Dale, Aug 28 2017 *)

PARI

a(n)=2*n+(n-1)^2\4*4 \\ Charles R Greathouse IV, May 21 2015

Python

a = lambda n: 2*n + 4*floor((n-1)**2/4)

Formula

a(n) = 2*n + 4*floor((n-1)^2/4) = 2*n + 4*A002620(n-1).

a(n) = A171218(n) - A171218(n-1). - Reinhard Zumkeller, Dec 05 2009

From R. J. Mathar, Jun 27 2011: (Start)

G.f.: 2*x*(1 + x^2) / ( (1 + x)*(1 - x)^3 ).

a(n) = 2*A000982(n). (End)

a(n+1) = (3 + 4*n + 2*n^2 + (-1)^n)/2 = A080335(n) + (-1)^n. - Philippe Deléham, Feb 17 2012

a(n) = 2 * ceiling(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

a(n) = n^2 + (n mod 2). - Bruno Berselli, Oct 03 2017

Sum_{n>=1} 1/a(n) = Pi*tanh(Pi/2)/4 + Pi^2/24. - Amiram Eldar, Jul 07 2022