

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.


9



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
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OFFSET

0,4


COMMENTS

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.


LINKS

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


FORMULA

a(2k+1) = a(2k) + 2*A000161(A001481(k+1))  A010052(A001481(k+1)/2).  Rajan Murthy, Jan 14 2013
a(2k) = a(2k1)  2*(A000161(A001481(k+1))  A010052(A001481(k+1))) + A010052(A001481(k+1)/2).  Rajan Murthy, Jan 14 2013


EXAMPLE

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.


PROG

(Scilab)
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 = l2
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


CROSSREFS

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 3dimensional Cartesian lattice and A239353 for the 4dimensional Cartesian lattice.
Sequence in context: A286244 A230010 A230847 * A181672 A125748 A058680
Adjacent sequences: A234297 A234298 A234299 * A234301 A234302 A234303


KEYWORD

nonn


AUTHOR

Rajan Murthy, Dec 22 2013


STATUS

approved



