A235141 -id:A235141 - cp's OEIS frontend

A235142 Numbers k such that A235141(k) = -1.

4, 10, 22, 32, 64, 84, 108, 132, 186, 214, 284, 360, 446, 490, 590, 642, 694, 746, 930, 990, 1192, 1266, 1342, 1568, 1738, 2086, 2180, 2276, 2470, 2572, 2668, 2780, 3326, 3556, 3680, 3922, 4298, 4430, 4560, 4832, 4968

Offset: 1

Rajan Murthy, Jan 03 2014

nonn

The positions reflect square radii which are uniquely twice a square integer, that is, the completion of only one square on the y = x line.

			For n = 2, a(2) = 10.  The 10th term of A235141 is -1 corresponding to the square radius of an origin centered circle increasing from the open interval(5,8) to exactly 8.

A000548(n) = (A001481(1 + a(n)/2 ) )/2.

Cf. A235141, A234300.

a(n) = A235386(n+1) - 1.

A235386 Numbers k such that A235141(k) = 1.

Original entry on oeis.org

1, 5, 11, 23, 33, 65, 85, 109, 133, 187, 215, 285, 361, 447, 491, 591, 643, 695, 747, 931, 991, 1193, 1267, 1343, 1569, 1739, 2087, 2181, 2277, 2471, 2573, 2669, 2781, 3327, 3557, 3681, 3923, 4299, 4431, 4561, 4833, 4969

Offset: 1

Rajan Murthy, Jan 08 2014

nonn

The positions reflect square radii which are uniquely twice a square integer, that is, the inclusion of only one square on the y = x line.

			for n=3, a(3) = 11. The eleventh term of A235141 is 1 reflecting an increase in the square radius of the circle from exactly 8 to the open interval of (8,9).

A000548(n) = A001481(1 + (a(n+1)-1)/2)/2.

Cf. A235141, A234300.

a(n+1) = A235142(n) + 1, a(1)=1.

A235143 Positions of -2 in A235141, the first differences of A234300.

Original entry on oeis.org

8, 14, 16, 20, 24, 26, 28, 30, 34, 38, 40, 42, 44, 50, 52, 54, 56, 62, 66, 68, 70, 74, 78, 80, 82, 86, 88, 90, 92, 94, 96, 98, 100, 104, 112, 114, 120, 122, 124, 126, 128, 130, 134, 136, 140, 142, 144, 146, 150, 152, 156, 160, 164, 166, 168, 172, 174, 176, 178, 180, 182, 184, 188, 190, 196, 200, 204

Offset: 1

Rajan Murthy, Jan 03 2014

nonn

The positions reflect radii which are a unique sum of two and only two distinct nonzero square integers.

The positions are a bit less frequent in occurrence than the positions where the first differences equal 2 because when the radius changes from exactly an integer value k to the open interval (k,k+1), the number of edge squares increase by 2, while in the reverse case, an increase from the open interval (k,k+1) to exactly k+1, the number of edge squares stays the same rather than decreasing by 2 as occurs in cases when the radii are a sum of two and only two distinct nonzero square integers. This is in contrast to positions where the first difference of A234300 equals 1 which are exactly balanced by positions which equal -1.

			a(1) = 8 which corresponds to the transition of the square radius from the interval (4,5) to 5 = 1^2 + 2^2.
a(2) = 14 which corresponds to the transition from (9,10) to 10 = 1^2 + 3^2.

Rajan Murthy, Table of n, a(n) for n = 1..1533

Cf. A235141, A234300, A235142, A235386.

A235387 Positions of 2's in A235141, the first differences of A234300.

Original entry on oeis.org

3, 7, 9, 13, 15, 17, 19, 21, 25, 29, 31, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 63, 67, 69, 71, 73, 75, 79, 81, 83, 89, 91, 93, 95, 97, 99, 101, 103, 105, 113, 115, 117, 121, 123, 125, 127, 129, 131, 135, 141, 143, 145, 147, 151, 153, 155, 157, 161, 165, 167, 169, 175, 177, 179, 181

Offset: 1

Rajan Murthy, Jan 08 2014

nonn

The positions reflect radii which are a unique sum of two distinct square integers where order doesn't matter.

The positions are more frequent in occurrence than the positions where the first differences equal -2 because when the radius changes from exactly an integer value k to the open interval (k,k+1), the number of edge squares increases by 2, while in the reverse case, an increase from the open interval (k,k+1) to exactly k+1, the number of edge squares stays the same. This is in contrast to positions where the first difference equals 1 which are exactly balanced by positions which equal -1 .

			a(2) = 7 corresponding to the shift from squared radius of 4 to (4,5).  This also marks a shift of the radius from 2 to (2,3).  The preceding shift, A235141(6), from radius in the interval (1,2) to 2 and squared radius in the interval (2,4) to 4 does not change the number of edge squares.
a(3) = 9 corresponding to the shift from squared radius of 5 to (5,8).  The radius however remains in the interval (2,3).  The preceding shift, A235141(8), from squared radius in the interval (4,5) to 5 results in a decrease of two due to the completion of the squares with upper right hand corner coordinates of x=1, y =2 and x=2, y=1 (since 5 = 1^2+2^2).

Rajan Murthy, Table of n, a(n) for n = 1..1553

Cf. A235141, A234300, A235142, A235386.

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

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A235142 Numbers k such that A235141(k) = -1.

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A235386 Numbers k such that A235141(k) = 1.

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A235143 Positions of -2 in A235141, the first differences of A234300.

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A235387 Positions of 2's in A235141, the first differences of A234300.

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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.

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