%I #18 Jun 02 2024 21:35:12

%S 1,5,21,37,61,89,129,177,221,277,341,401,489,561,657,749,845,949,1049,

%T 1185,1313,1441,1573,1709,1877,2025,2185,2361,2529,2709,2901,3101,

%U 3305,3505,3713,3917,4157,4397,4637,4865,5121,5377,5637,5917,6197,6485,6761

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

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

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

%C Symmetrical about the horizontal and vertical axes.

%H David Dewan, <a href="/A373193/b373193.txt">Table of n, a(n) for n = 1..10000</a>

%H David Dewan, <a href="/A373193/a373193.pdf">Drawings for n=1..10</a>

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

%e For n=4:

%e row 1: 3 squares - - X X X - -

%e row 2: 5 squares - X X X X X -

%e row 3: 7 squares X X X X X X X

%e row 4: 7 squares X X X X X X X

%e row 5: 7 squares X X X X X X X

%e row 6: 5 squares - X X X X X -

%e row 7: 3 squares - - X X X - -

%e Total = 37 = a(4).

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

%Y 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).

%K nonn

%O 1,2

%A _David Dewan_, May 27 2024