cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A003881, A136483, A136513.
Alternating merge of A119677 of A136485.

Programs

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

Formula

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

A136513 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter n.

Original entry on oeis.org

0, 0, 2, 2, 6, 8, 12, 16, 26, 30, 38, 44, 56, 60, 74, 82, 96, 108, 128, 138, 154, 166, 188, 196, 220, 238, 262, 278, 304, 324, 344, 366, 398, 416, 452, 468, 506, 526, 562, 588, 616, 644, 686, 714, 754, 780, 824, 848, 894, 930, 976, 1008, 1056, 1090, 1134, 1170
Offset: 1

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

Alternating merge of A136514 and A136515.
Cf. A136483, A136485.

Programs

  • Magma
    A136513:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136513(n): n in [1..100]]; // G. C. Greubel, Jul 27 2023
  • Mathematica
    Table[2*Sum[Floor[Sqrt[(n/2)^2 -k^2]], {k,Floor[n/2]}], {n,100}]
  • PARI
    a(n) = 2*sum(k=1, n\2, sqrtint((n/2)^2-k^2)); \\ Michel Marcus, Jul 27 2023
  • SageMath
    def A136513(n): return 2*sum(isqrt((n/2)^2-k^2) for k in range(1,(n//2)+1))
[A136513(n) for n in range(1,101)] # G. C. Greubel, Jul 27 2023

Formula

Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
a(n) = 2 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
a(n) = 2 * A136483(n).
a(n) = (1/2) * A136485(n).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (2 * (1 - x)). - Ilya Gutkovskiy, Nov 24 2021

A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

Original entry on oeis.org

0, 1, 3, 6, 13, 19, 28, 37, 48, 64, 77, 94, 110, 131, 152, 172, 199, 226, 253, 281, 308, 343, 377, 412, 447, 488, 528, 567, 612, 654, 703, 750, 796, 847, 902, 957, 1013, 1068, 1129, 1187, 1252, 1313, 1378, 1446, 1511, 1582, 1650, 1725, 1800, 1877, 1955, 2034
Offset: 0

Views

Author

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

Keywords

Comments

Number of unit square lattice cells inside quadrant of origin centered circle of radius n+1/2.

Examples

			a(2) = 3 because a circle of radius 2+1/2 in the first quadrant encloses (2,1), (1,1), (1,2).

Links

Crossrefs

Cf. A019683, A136483, A136486, A136515.

Programs

  • Magma
    A136484:= func< n | n eq 0 select 0 else (&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
[A136484(n): n in [0..100]]; // G. C. Greubel, Jul 29 2023
  • Mathematica
    Table[Sum[Floor[Sqrt[(n+1/2)^2 - k^2]], {k,n}], {n,0,100}]
  • SageMath
    def A136484(n): return sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
[A136484(n) for n in range(101)] # G. C. Greubel, Jul 29 2023

Formula

a(n) = Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
a(n) = (1/2) * A136515(n).
a(n) = (1/4) * A136486(n).
a(n) = A136483(2*n+1).
Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).

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

Views

Author

David Dewan, May 14 2024

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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).

Programs

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

Formula

a(n) = 2*Sum_{k=1..n-1} floor(2*sqrt(n^2 - k^2)).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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