A111746 Number of squares in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

0, 5, 17, 42, 82, 143, 227, 340, 484, 665, 885, 1150, 1462, 1827, 2247, 2728, 3272, 3885, 4569, 5330, 6170, 7095, 8107, 9212, 10412, 11713, 13117, 14630, 16254, 17995, 19855, 21840, 23952, 26197, 28577, 31098, 33762, 36575, 39539, 42660, 45940

Offset: 0

Author

Floor van Lamoen, Nov 19 2005

Keywords

• nonn
• easy

Crossrefs

Cf. A000330, A000447, A002623.

Programs

Magma

[n*(4*n^2+12*n+11)/6+1/4-(-1)^n/4: n in [0..60]]; // Vincenzo Librandi, May 12 2015

Maple

seq(n*(4*n^2-1)/6 - 1/4 + 1/4*(-1)^n,n=1..50);

Mathematica

Table[n (4 n^2 + 12 n + 11)/6 + 1/4 - (-1)^n/4, {n, 0, 60}] (* Vincenzo Librandi, May 12 2015 *)

Formula

a(n) = n*(4*n^2 + 12*n + 11)/6 + 1/4 - (-1)^n/4 = floor(A000447(n+1)/2).

a(n) = 4*A002623(n-1) + A000330(n), with A002623(-1)=0. - Luce ETIENNE, May 12 2015

G.f.: x*(5 + 2*x + x^2)/((1-x)^4*(1+x)). - Vincenzo Librandi, May 12 2015

Extensions

Closed formula adapted to the offset by Bruno Berselli, May 12 2015