A373193 On a unit square grid, the number of squares enclosed by a circle of radius n with origin at the center of a square.

1, 5, 21, 37, 61, 89, 129, 177, 221, 277, 341, 401, 489, 561, 657, 749, 845, 949, 1049, 1185, 1313, 1441, 1573, 1709, 1877, 2025, 2185, 2361, 2529, 2709, 2901, 3101, 3305, 3505, 3713, 3917, 4157, 4397, 4637, 4865, 5121, 5377, 5637, 5917, 6197, 6485, 6761

Offset: 1

Author

David Dewan, May 27 2024

Keywords

• nonn

Comments

This corresponds to a circle of radius n with center at 1/2,1/2 on a unit square grid.

Always has an odd number of rows (2 n - 1) with an odd number of squares in each row.

Symmetrical about the horizontal and vertical axes.

Examples

			For n=4:
  row 1: 3 squares   - - X X X - -
  row 2: 5 squares   - X X X X X -
  row 3: 7 squares   X X X X X X X
  row 4: 7 squares   X X X X X X X
  row 5: 7 squares   X X X X X X X
  row 6: 5 squares   - X X X X X -
  row 7: 3 squares   - - X X X - -
Total = 37 = a(4).
		

Links

• David Dewan, Table of n, a(n) for n = 1..10000
• David Dewan, Drawings for n=1..10

Crossrefs

Cf. A119677 (on unit square grid with circle center at origin), A372847 (even number of rows with maximal squares per row), A125228 (odd number of rows with maximal squares per row), A000328 (number of squares whose centers are inside the circle).

Programs

Mathematica

Table[4*Sum[Floor[Sqrt[n^2-(k+1/2)^2]-1/2],{k,1,n-1}]+4*n-3,{n,50}]

Formula

a(n) = 4*Sum_{k=1..n-1} floor(sqrt(n^2 - (k+1/2)^2) - 1/2) + 4*n - 3.

a(n) == 1 (mod 4). - Robert FERREOL, Jan 31 2025