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Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.
3

%I #19 May 16 2024 20:34:15

%S 0,6,18,36,64,92,130,172,224,284,344,410,488,570,658,750,852,956,1072,

%T 1194,1312,1450,1584,1728,1882,2044,2204,2372,2548,2730,2916,3112,

%U 3312,3520,3738,3950,4184,4408,4656,4900,5146,5402,5670,5942,6222,6492,6784,7080,7382,7700

%N Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.

%C Always has an even number of rows (2*n-2) and each row may have an odd or even number of squares.

%C Symmetrical about the horizontal and vertical axes.

%H David Dewan, <a href="/A372847/b372847.txt">Table of n, a(n) for n = 1..10000</a>

%H David Dewan, <a href="/A372847/a372847.pdf">Drawings for n=1..12.</a>

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

%e For n=4

%e row 1: 5 squares

%e row 2: 6 squares

%e row 3: 7 squares

%e row 4: 7 squares

%e row 5: 6 squares

%e row 6: 5 squares

%e Total = 36

%t a[n_]:=2 Sum[Floor[2 Sqrt[n^2 - k^2]], {k,n-1}]; Array[a,50]

%Y Cf. A136485 (by diameter), A001182 (within quadrant), A136483 (quadrant by diameter), A119677 (even number of rows with even number of squares in each), A125228 (odd number of rows with maximal squares per row), A341198 (points rather than squares).

%K nonn

%O 1,2

%A _David Dewan_, May 14 2024