A124623 Number of unit squares having center within inscribed circle of an n X n integer square.

1, 4, 9, 12, 21, 32, 37, 52, 69, 80, 97, 112, 137, 156, 177, 208, 225, 256, 293, 316, 349, 384, 421, 448, 489, 540, 577, 616, 665, 716, 749, 812, 861, 912, 973, 1020, 1085, 1124, 1201, 1264, 1313, 1396, 1457, 1528, 1597, 1664, 1741, 1804, 1885, 1976, 2053, 2128

Offset: 1

Author

William A. Berry (waberr2(AT)uky.edu), Dec 21 2006

Keywords

• easy
• nonn

Comments

From Robert G. Wilson v, Mar 22 2017: (Start)

For n odd, the center of the circle is in the middle of the center square and thus a(2n-1) == 1 (mod 4).

For n even, the center of the circle is at the four corners of the center 4 squares and thus a(2n) == 0 (mod 4).

a(n) ~ n*Pi/4. (End)

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051233.

Programs

Mathematica

f[n_] := 4*Length[ Select[ Flatten[ Table[ If[ OddQ@ n, x^2 + y^2, x(x -1) + y(y -1) + 1/2], {x, n/2}, {y, n/2}]], 4# < n^2 &]] + If[ OddQ@ n, 2(n -1) + 1, 0]; Array[f, 52] (* Robert G. Wilson v, Mar 22 2017 *)

Formula

a(n) = n^2 - 4*k(n); k(n) = number of exterior centers per quadrant.

a(2n-1) = A036704(n-1). - Robert G. Wilson v, Mar 28 2017

a(2n) = 4*A120883(n-1). - Robert G. Wilson v, Mar 28 2017