OFFSET
1,1
COMMENTS
This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = 2*n + 4*floor((n-1)^2/4) = 2*n + 4*A002620(n-1).
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
EXAMPLE
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.
MAPLE
MATHEMATICA
LinearRecurrence[{2, 0, -2, 1}, {2, 4, 10, 16}, 60] (* Harvey P. Dale, Aug 28 2017 *)
PROG
(Python) a = lambda n: 2*n + 4*floor((n-1)**2/4)
(PARI) a(n)=2*n+(n-1)^2\4*4 \\ Charles R Greathouse IV, May 21 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
William A. Tedeschi, Feb 29 2008
STATUS
approved