A119677 -id:A119677 - cp's OEIS frontend

A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

0, 0, 4, 4, 12, 16, 24, 32, 52, 60, 76, 88, 112, 120, 148, 164, 192, 216, 256, 276, 308, 332, 376, 392, 440, 476, 524, 556, 608, 648, 688, 732, 796, 832, 904, 936, 1012, 1052, 1124, 1176, 1232, 1288, 1372, 1428, 1508, 1560, 1648, 1696, 1788, 1860, 1952, 2016

Offset: 1

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

a(n) is the number of complete squares that fit inside the circle with diameter n, drawn on squared paper.

			a(3) = 4 because a circle centered at the origin and of radius 3/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003881, A136483, A136513.

Alternating merge of A119677 of A136485.

A136485:= func< n | n le 1 select 0 else 4*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136485(n): n in [1..100]]; // G. C. Greubel, Jul 29 2023

Table[4*Sum[Floor[Sqrt[(n/2)^2 - k^2]], {k,Floor[n/2]}], {n,100}]

def A136485(n): return 4*sum(floor(sqrt((n/2)^2-k^2)) for k in range(1,(n//2)+1))
[A136485(n) for n in range(1,101)] # G. C. Greubel, Jul 29 2023

a(n) = 4 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
a(n) = 4 * A136483(n).
a(n) = 2 * A136513(n).
Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (1 - x). - Ilya Gutkovskiy, Nov 24 2021

A372847 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.

Original entry on oeis.org

0, 6, 18, 36, 64, 92, 130, 172, 224, 284, 344, 410, 488, 570, 658, 750, 852, 956, 1072, 1194, 1312, 1450, 1584, 1728, 1882, 2044, 2204, 2372, 2548, 2730, 2916, 3112, 3312, 3520, 3738, 3950, 4184, 4408, 4656, 4900, 5146, 5402, 5670, 5942, 6222, 6492, 6784, 7080, 7382, 7700

Offset: 1

David Dewan, May 14 2024

nonn

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

Symmetrical about the horizontal and vertical axes.

			For n=4
row 1:   5 squares
row 2:   6 squares
row 3:   7 squares
row 4:   7 squares
row 5:   6 squares
row 6:   5 squares
Total = 36

David Dewan, Table of n, a(n) for n = 1..10000
David Dewan, Drawings for n=1..12.

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).

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

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

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.

Original entry on oeis.org

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

David Dewan, May 27 2024

nonn

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.

			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).

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

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).

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

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

A374532 Number of complete unit squares that fit inside a circle of radius sqrt(n^2+1) centered at the origin of a square lattice.

Original entry on oeis.org

0, 4, 12, 24, 40, 68, 96, 132, 180, 224, 284, 340, 408, 492, 564, 656, 740, 848, 960, 1060, 1184, 1304, 1444, 1576, 1704, 1868, 2024, 2196, 2356, 2520, 2716, 2892, 3104, 3292, 3504, 3720, 3916, 4160, 4384, 4628, 4872, 5108, 5372, 5640, 5916, 6188, 6456, 6764, 7036

Offset: 0

Thomas Otten, Jul 10 2024

nonn

Thomas Otten, Illustration of initial terms.

Cf. A119677 (case for radius of n), A237526.

Cf. A046092, A000328 (quadrant width 1 cell).

a(n) = my(s=n^2+1); 4*sum(k=1, sqrtint(s), sqrtint(s-k^2)) \\ Andrew Howroyd, Jul 11 2024

def A374532(n): return sum(isqrt(k*((n<<1)-k)+1) for k in range(n))<<2 # Chai Wah Wu, Jul 18 2024

a(n) = 4*A237526(n^2 + 1).

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A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

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A372847 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.

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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.

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A374532 Number of complete unit squares that fit inside a circle of radius sqrt(n^2+1) centered at the origin of a square lattice.

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