cp's OEIS Frontend

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.

%I A234300 #31 Apr 28 2014 14:56:26
%S A234300 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,
%T A234300 10,11,9,11,11,13,11,13,11,13,11,13,11,13,13,15,12,15,13,15,13,15,13,
%U A234300 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
%N 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.
%C A234300 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.
%C A234300 a(n) is even when the radius squared corresponds to an element of A024517.
%H A234300 Rajan Murthy, <a href="/A234300/b234300.txt">Table of n, a(n) for n = 0..4999</a>
%F A234300 a(2k+1) = a(2k) + 2*A000161(A001481(k+1)) - A010052(A001481(k+1)/2). - _Rajan Murthy_, Jan 14 2013
%F A234300 a(2k) = a(2k-1) - 2*(A000161(A001481(k+1)) - A010052(A001481(k+1))) + A010052(A001481(k+1)/2). - _Rajan Murthy_, Jan 14 2013
%e A234300 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.
%o A234300 (Scilab)
%o A234300 function index = n_edgeindex (N)
%o A234300     if N < 1 then
%o A234300         N = 1
%o A234300     end
%o A234300     N = floor(N)
%o A234300     i = 0:ceil(N/2)
%o A234300     i = i^2
%o A234300     index = i
%o A234300     for j = 1:length(i)
%o A234300        index = [index i+ i(j).*ones(i)]
%o A234300     end
%o A234300     index = unique(index)
%o A234300     index = index(1:ceil(N))
%o A234300     d = diff(index)/2
%o A234300     d = d +  index(1:length(d))
%o A234300     index = gsort([index d],"g","i")
%o A234300     index = index(1:N)
%o A234300 endfunction
%o A234300 function l = n_edge_n (i)
%o A234300         l=0
%o A234300         h=0
%o A234300         while (i > (2*h^2))
%o A234300             h=h+1
%o A234300         end
%o A234300         if i < (2*h^2) then
%o A234300                 l = l+1
%o A234300         end
%o A234300         if i >1 then
%o A234300             t=[0 1]
%o A234300            while (i>max(t))
%o A234300                t = [t (sqrt(max(t))+1)^2]
%o A234300            end
%o A234300         for j = 1:h
%o A234300            b=t
%o A234300            t=[2*(j)^2 (j+1)^2 + (j)^2]
%o A234300            while (i>max(t))
%o A234300                t = [t (sqrt(max(t)-(j)^2)+1)^2 + (j)^2]
%o A234300            end
%o A234300            l = l+ 2*(length(b)-length(t))
%o A234300            if max(t) == i then
%o A234300                l = l-2
%o A234300            end
%o A234300         end
%o A234300        end
%o A234300 endfunction
%o A234300 function a =n_edge (N)
%o A234300     if N <1 then
%o A234300         N =1
%o A234300     end
%o A234300     N = floor(N)
%o A234300     a= []
%o A234300     index = n_edgeindex(N)
%o A234300     for i = index
%o A234300         a = [a n_edge_n(i)]
%o A234300     end
%o A234300 endfunction
%Y A234300 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.
%Y A234300 A237708 is the analog for the 3-dimensional Cartesian lattice and A239353 for the 4-dimensional Cartesian lattice.
%K A234300 nonn
%O A234300 0,4
%A A234300 _Rajan Murthy_, Dec 22 2013