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Maximal number of squares of side 1 in a disk of radius n.
5

%I #18 Jul 19 2024 14:31:00

%S 1,7,21,39,65,93,135,179,227,285,349,415,495,573,663,759,859,963,1071,

%T 1199,1325,1457,1591,1735,1891,2049,2207,2383,2557,2735,2929,3123,

%U 3327,3529,3739,3955,4191,4427,4665,4901,5159,5413,5681,5951,6231,6515,6799

%N Maximal number of squares of side 1 in a disk of radius n.

%H David Dewan, <a href="/A125228/b125228.txt">Table of n, a(n) for n = 1..10000</a>

%H David Dewan, <a href="/A125228/a125228.pdf">Drawings for n=1..12</a>

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

%e a(2)=7 since you cannot pack more than 7 unit-side squares in a disk of radius 2

%t f[n_] := 2 Sum[ IntegerPart[2 Sqrt[n^2 - (n - k - 1/2)^2]], {k, 0, n - 2}] + IntegerPart[2 Sqrt[n^2 - 1/2^2]]; Array[f, 47] (* _Robert G. Wilson v_, Jan 27 2007 *)

%t a[n_]:=2 Sum[Floor[2 Sqrt[n^2-(k+1/2)^2]],{k,n-1}]+2n-1;

%t Array[a, 47] (* _David Dewan_, Jun 07 2024*)

%o (Python)

%o from math import isqrt

%o def A125228(n): return (m:=n<<1)-1+(sum(isqrt((k*(m-k+1)-n<<2)-1) for k in range(1,n))<<1) # _Chai Wah Wu_, Jul 18 2024

%Y Similar to A001182 but less constrained.

%Y A124484 is another version.

%K easy,nonn

%O 1,2

%A Filippo ALUFFI PENTINI (falpen(AT)gmail.com), Jan 25 2007

%E More terms from _Robert G. Wilson v_, Jan 27 2007