A237707 -id:A237707 - cp's OEIS frontend

A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 45, 47, 48, 50, 52, 54, 56, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 90, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 115, 117, 119, 123, 125, 127, 129, 131

Offset: 1

Rajan Murthy and Vale Murthy, Nov 24 2013

nonn

The interval between terms reflects the number of ways a square integer can be partitioned into the sum of two square integers in an ordered pair. As examples, the increase from a(1) to a(2) from 1 to 3 is due to the inclusion of (1,2) and (2,1); and the increase from a(2) to a(3) is due to the inclusion of (2,2). Larger intervals occur when there are more combinations, such as, between a(17) and a(18) when (1,7), (7,1), and (5,5) are included.

			When radius of the circle exceeds 2^(1/2), one square is completely within the circle until the radius reaches 5^(1/2) when three squares are completely within the circle.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..2623 from Rajan Murthy).
L. Edson Jeffery, Illustration of the first few terms.
Rajan Murthy, Graph of sequence
Rajan Murthy, Graph of intervals
Rajan Murthy, Diagram depicting A(13)

First differences are in A229904.
The first differences must be odd at positions given in A024517 by proof by symmetry as r^2=2*n^2 is on the x=y line.
The radii corresponding to the terms are given by the square roots of A000404.
Cf. A001481, A057961.
Cf. A237707 (3-dimensional analog), A239353 (4-dimensional analog).

(* An empirical solution *) terms = 100; f[r_] := Sum[Floor[Sqrt[r^2 - n^2]], {n, 1, Floor[r]}]; Clear[g]; g[m_] := g[m] = Union[Table[f[Sqrt[s]], {s, 2, m }]][[1 ;; terms]]; g[m = dm = 4*terms]; g[m = m + dm]; While[g[m] != g[m - dm], Print[m]; m = m + dm]; A232499 = g[m]  (* Jean-François Alcover, Mar 06 2014 *)

A014465 A063691 without zeros.

Original entry on oeis.org

1, 3, 3, 3, 1, 6, 3, 3, 3, 6, 3, 3, 6, 4, 6, 6, 6, 3, 6, 3, 9, 9, 6, 3, 3, 6, 6, 1, 6, 6, 6, 6, 12, 6, 6, 9, 6, 12, 6, 12, 3, 3, 12, 6, 3, 3, 12, 7, 3, 12, 6, 12, 3, 9, 6, 15, 3, 15, 12, 6, 6, 12, 3, 3, 12, 9, 18, 6, 6, 12, 6, 9, 4, 6, 18, 9, 12, 6, 6, 12, 9, 6, 9, 12, 6, 12, 18, 18, 15, 6, 6, 21, 3, 9, 12, 9, 6, 12

Offset: 1

A. Timothy Royappa, 1997; entry revised Jun 13 2003

Let b(n) = n-th number of form x^2 + y^2 + z^2, x,y,z >= 1 (A000408); a(n) = number of solutions (x,y,z) to x^2 + y^2 + z^2 = b(n).
The a(n) are also the degeneracies of the energy levels E(n) in the three-dimensional cubic "particle-in-a-box" model in elementary quantum mechanics. - A. Timothy Royappa, Jan 09 2009
Continuously increase the radius of a sphere centered at the origin. Whenever the number of entire unit cubes that fit into one quadrant of the sphere increases (cf. A237707), list the number of additional cubes. - M. F. Hasler, Jun 25 2022

			b(1) = 3 = 1^2 + 1^2 + 1^2 (1 way), so a(1) = 1;
b(2) = 6 = 2^2 + 1^2 + 1^2 (3 ways), so a(2) = 3; etc.

G. M. Barrow, Physical Chemistry (6th ed.), McGraw-Hill, 1996, p. 69.

Daniel Leary, Table of n, a(n) for n = 1..8283

First différences of A237707.

for(n=1,200,r=sqrtint(n);s=0;for(i=1,r,si=i*i;for(j=1,r,sj=j*j;for(k=1,r,if(si+sj+k*k==n,s=s+1))));if(s,print1(s,","))) /* Ralf Stephan, Aug 31 2013 */

More terms and better name from Ralf Stephan, Aug 31 2013

A239353 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, completely within the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 5, 11, 15, 19, 31, 32, 44, 48, 54, 58, 70, 82, 94, 100, 112, 124, 148, 164, 176, 194, 206, 219, 235, 247, 275, 281, 317, 333, 345, 369, 393, 417, 421, 437

Offset: 1

Rajan Murthy, Mar 16 2014

nonn

			When the radius of the sphere reaches 2, one cube is completely within the sphere. When the radius reaches 7^(1/2), five cubes are completely within the sphere.

Rajan Murthy, Table of n, a(n) for n = 1..100
Rajan Murthy, Scilab code for generating the sequence

Cf. A237707 (3-dimensional analog), A232499 (2-dimensional analog). The square radii corresponding to the elements of {a(n)} are the indices of the nonzero terms of A025428.

A239318 Number of visible unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 3, 6, 6, 7, 10, 12, 12, 15, 15, 18, 18, 21, 22, 25, 25, 28, 30, 30, 33, 33, 36, 39, 39, 39, 42, 42, 43, 43, 46, 49, 49, 52, 52, 54, 57, 57, 60, 63, 66, 66, 66, 66, 69, 69, 72, 72, 73, 76, 79, 79, 82, 82, 82, 82, 87, 90, 93, 93, 93, 93, 96, 96, 99, 99, 102

Offset: 1

Frans Faase, Mar 15 2014

nonn

This sequence is related to A237707, which gives all unit cubes. For each element in that sequence, this sequence gives the number of cubes that have at least one side that is visible from the sphere.

Cf. A237707.

void printVisibleCubes(int nr)
{
    int prev_c = 0;
    for (int r = 3; nr > 0; r++)
    {
        int c = 0;
        int c2 = 0;
        for (int i = 1; i*i + 2 <= r; i++)
            for (int j = 1; i*i + j*j + 1 <= r; j++)
                for (int k = 1; i*i + j*j + k*k <= r; k++)
                {
                    c++;
                    if (   (i+1)*(i+1) + j*j + k*k > r
                        || i*i + (j+1)*(j+1) + k*k > r
                        || i*i + j*j + (k+1)*(k+1) > r)
                        c2++;
                }
        if (c > prev_c)
        {
            // c contains next element of A237707
            printf("%d, ", c2);
            prev_c = c;
            nr--;
        }
    }
}

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A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

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A014465 A063691 without zeros.

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A239353 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, completely within the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

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A239318 Number of visible unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

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C