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

1,2

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

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

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 *)

Crossrefs

Cf. A051233.

Sequence in context: A332225 A186130 A051233 * A220195 A221355 A197681

Adjacent sequences: A124620 A124621 A124622 * A124624 A124625 A124626

Keyword

easy,nonn

Author

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

Status

approved