A051233 -id:A051233 - cp's OEIS frontend

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

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

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)

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

Cf. A051233.

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 *)

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

A033551 Closest integer to (Pi/4)*n^2.

Original entry on oeis.org

1, 3, 7, 13, 20, 28, 38, 50, 64, 79, 95, 113, 133, 154, 177, 201, 227, 254, 284, 314, 346, 380, 415, 452, 491, 531, 573, 616, 661, 707, 755, 804, 855, 908, 962, 1018, 1075, 1134, 1195, 1257, 1320, 1385, 1452, 1521

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

			a(3)=7, since 3^2*Pi/4 = 7.06858347.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Approximation for A051233.

List([1..50], n-> Int(Round(Atan(1.0)*n^2)) ); # G. C. Greubel, Oct 12 2019

R:= RealField(20); [Round(Pi(R)*n^2/4): n in [1..50]]; // G. C. Greubel, Oct 12 2019

seq(round((1/4)*Pi*n^2), n = 1..50); # G. C. Greubel, Oct 12 2019

Round[Pi/4 Range[50]^2] (* Harvey P. Dale, May 11 2016 *)

a(n) = round((Pi/4) * n^2); \\ Michel Marcus, Sep 02 2013

[round(pi*n^2/4) for n in (1..50)] # G. C. Greubel, Oct 12 2019

a(n) = round( (Pi/4) * n^2 ).

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

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A033551 Closest integer to (Pi/4)*n^2.

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