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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.

%I A373193 #29 Feb 09 2025 15:47:16
%S A373193 1,5,21,37,61,89,129,177,221,277,341,401,489,561,657,749,845,949,1049,
%T A373193 1185,1313,1441,1573,1709,1877,2025,2185,2361,2529,2709,2901,3101,
%U A373193 3305,3505,3713,3917,4157,4397,4637,4865,5121,5377,5637,5917,6197,6485,6761
%N 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.
%C A373193 This corresponds to a circle of radius n with center at 1/2,1/2 on a unit square grid.
%C A373193 Always has an odd number of rows (2 n - 1) with an odd number of squares in each row.
%C A373193 Symmetrical about the horizontal and vertical axes.
%H A373193 David Dewan, <a href="/A373193/b373193.txt">Table of n, a(n) for n = 1..10000</a>
%H A373193 David Dewan, <a href="/A373193/a373193.pdf">Drawings for n=1..10</a>
%F A373193 a(n) = 4*Sum_{k=1..n-1} floor(sqrt(n^2 - (k+1/2)^2) - 1/2) + 4*n - 3.
%F A373193 a(n) == 1 (mod 4). - _Robert FERREOL_, Jan 31 2025
%e A373193 For n=4:
%e A373193   row 1: 3 squares   - - X X X - -
%e A373193   row 2: 5 squares   - X X X X X -
%e A373193   row 3: 7 squares   X X X X X X X
%e A373193   row 4: 7 squares   X X X X X X X
%e A373193   row 5: 7 squares   X X X X X X X
%e A373193   row 6: 5 squares   - X X X X X -
%e A373193   row 7: 3 squares   - - X X X - -
%e A373193 Total = 37 = a(4).
%t A373193 Table[4*Sum[Floor[Sqrt[n^2-(k+1/2)^2]-1/2],{k,1,n-1}]+4*n-3,{n,50}]
%Y A373193 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 A373193 nonn
%O A373193 1,2
%A A373193 _David Dewan_, May 27 2024