A239353 -id:A239353 - 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 *)

A234300 Number of unit squares, aligned with a Cartesian grid, partially encircled along the edge of the first quadrant of a circle centered at the origin ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 5, 3, 5, 4, 5, 5, 7, 5, 7, 5, 7, 7, 9, 7, 9, 8, 9, 7, 9, 7, 11, 9, 11, 9, 11, 10, 11, 9, 11, 11, 13, 11, 13, 11, 13, 11, 13, 11, 13, 13, 15, 12, 15, 13, 15, 13, 15, 13, 15, 13, 15, 15, 17, 13, 17, 15, 17, 16, 17, 15, 17, 15, 17, 15, 17, 17, 19, 17, 19, 15, 19, 17, 19, 17, 19, 17, 19, 18, 19, 17, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21, 19, 21

Offset: 0

Rajan Murthy, Dec 22 2013

nonn

The first decrease from a(4) = 3 to a(5) = 2 occurs when the radius squared increases from an arbitrary position between 1 and 2 (when 3 squares are on the edge) to exactly 2 (when only 2 squares are on the edge because the circle of square radius 2 passes through the upper right corner on the y=x line). Similar decreases occur when the circle passes through other upper right corners. At least some (if not all) adjacent duplicates occur when the square radius corresponds to a perfect square, that is a corner which is only a lower right corner, i.e., on the y = 0 line. For example, a(6)=a(7)=3 occurs when, for n = 6 , a(n) corresponding to the interval between 2 and 4; and, for n=7, a(n) corresponding to the exact square radius of 4. Some of the confusion may come from the fact that for odd n, there is a unique circle corresponding to elements of a(n) (passing through the corner of specific square(s) on the grid), while for even n, there is a set of circles with a range of radii (which do not pass through corners) corresponding to the elements of a(n). It seems easier to organize the concept in terms of intervals and corners for the sake of consistency.

a(n) is even when the radius squared corresponds to an element of A024517.

			At radius 0, there are no partially filled squares.  At radius >1 but < sqrt(2), there are 3 partially filled squares along the edge of the circle.  At radius = sqrt(2), there are 2 partially filled squares along the edge of the circle.

Rajan Murthy, Table of n, a(n) for n = 0..4999

Cf. A001481 (corresponds to the square radius of alternate entries), A232499 (number of completely encircled squares when the radii are indexed by A000404), A235141 (first differences), A024517.

A237708 is the analog for the 3-dimensional Cartesian lattice and A239353 for the 4-dimensional Cartesian lattice.

function index = n_edgeindex (N)
    if N < 1 then
        N = 1
    end
    N = floor(N)
    i = 0:ceil(N/2)
    i = i^2
    index = i
    for j = 1:length(i)
       index = [index i+ i(j).*ones(i)]
    end
    index = unique(index)
    index = index(1:ceil(N))
    d = diff(index)/2
    d = d +  index(1:length(d))
    index = gsort([index d],"g","i")
    index = index(1:N)
endfunction
function l = n_edge_n (i)
        l=0
        h=0
        while (i > (2*h^2))
            h=h+1
        end
        if i < (2*h^2) then
                l = l+1
        end
        if i >1 then
            t=[0 1]
           while (i>max(t))
               t = [t (sqrt(max(t))+1)^2]
           end
        for j = 1:h
           b=t
           t=[2*(j)^2 (j+1)^2 + (j)^2]
           while (i>max(t))
               t = [t (sqrt(max(t)-(j)^2)+1)^2 + (j)^2]
           end
           l = l+ 2*(length(b)-length(t))
           if max(t) == i then
               l = l-2
           end
        end
       end
endfunction
function a =n_edge (N)
    if N <1 then
        N =1
    end
    N = floor(N)
    a= []
    index = n_edgeindex(N)
    for i = index
        a = [a n_edge_n(i)]
    end
endfunction

a(2k+1) = a(2k) + 2*A000161(A001481(k+1)) - A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

a(2k) = a(2k-1) - 2*(A000161(A001481(k+1)) - A010052(A001481(k+1))) + A010052(A001481(k+1)/2). - Rajan Murthy, Jan 14 2013

A240600 Number of partially filled hexagons in the first 120-degree circular sector of hexagonal lattice A_2 centered at deep hole along the edge of a circle also centered at the deep hole.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 3, 3, 5, 4, 5, 5, 7, 5, 5, 5, 6, 6, 8, 6, 8, 7, 7, 7, 9, 7, 7, 7, 9, 8, 9, 9, 11, 9, 11, 9, 9, 9, 11, 9, 10, 10, 12, 10, 12, 12, 14, 12, 14, 13, 13, 11, 11, 11, 13, 13, 15, 13, 13, 13, 15, 14

Offset: 0

Rajan Murthy, Apr 09 2014

nonn

A(n) alternates between the numbers for circles which intersect points on the A2 lattice and the numbers for circles which pass in between the points on a lattice.

			for n = 1, the squared radius is in the open interval (0,1) and the corresponding circle passes through 1 hexagon.
for n = 14, the squared radius is 13 with the corresponding circle passing through the furthest corner of 2 hexagons and passing through 5 hexagons.

Rajan Murthy, Code for computing 3*A(n) as a function of R^2 (scilab)

A038588 gives the number of hexagons completely encircled in all three circular sectors.
Squared radii of alternate entries is given by the Loeschian numbers A003136.
A234300 is the analog for the 2-d Cartesian lattice.
A237708 is the analog for the 3-d Cartesian lattice.
A239353 is the analog for the 4-d Cartesian lattice.

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A234300 Number of unit squares, aligned with a Cartesian grid, partially encircled along the edge of the first quadrant of a circle centered at the origin ordered by increasing radius.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scilab

Formula

A240600 Number of partially filled hexagons in the first 120-degree circular sector of hexagonal lattice A_2 centered at deep hole along the edge of a circle also centered at the deep hole.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs