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%I #17 May 01 2013 16:06:53
%S 0,2,2,0,3,3,3,5,0,4,4,6,6,4,4,0,7,7,6,8,7,6,5,8,0,6,8,10,9,6,7,11,10,
%T 8,6,0,11,7,7,12,12,7,9,11,13,8,7,13,0,10,9,16,12,12,9,12,14,10,9,12,
%U 15,8,11,0,15,10,11,16,13,11,9,19,18,10,11,18,13
%N Total number of solutions to the equation x^2 + k*y^2 = n with x >= 0, y >= 0, k > 0, or 0 if the number is infinite. (Order does not matter for the equation x^2 + y^2 = n).
%C If the equation x^2+y^2 = n has got two solutions (x, y), (y, x) then they will be counted only once.
%C No solutions can exist for the values of k > n.
%C This sequence differs from A216504, since this sequence gives the total number of solutions to the equation x^2+k*y^2 = n, whereas the sequence A216504 gives the number of distinct values of k for which a solution to the equation x^2+k*y^2 = n can exist.
%C Some values of k can clearly have more than one solution.
%C For example, x^2+k*y^2 = 33 is satisfiable for
%C 33 = 1^2+2*4^2.
%C 33 = 5^2+2*2^2.
%C 33 = 3^2+6*2^2.
%C 33 = 1^2+8*2^2.
%C 33 = 5^2+8*1^2.
%C 33 = 4^2+17*1^2.
%C 33 = 3^2+24*1^2.
%C 33 = 2^2+29*1^2.
%C 33 = 1^2+32*1^2.
%C 33 = 0^2+33*1^2.
%C So for this sequence a(33) = 10.
%C On the other hand, for A216504, there exist only 7 different values of k for which a solution to the equation mentioned above exists.
%C So A216504(33) = 8.
%o (PARI) for(n=1, 100, sol=0; for(k=1, n, for(x=0, n, if((issquare(n-k*x*x)&&n-k*x*x>=0&&k>=2)||(issquare(n-x*x)&&n-x*x>=0&&k==1&&x*x<=n-x*x), sol++))); if(issquare(n),print1(0", "),print1(sol", "))) /* _V. Raman_, Oct 16 2012 */
%Y Cf. A216503, A216504, A216505.
%Y Cf. A217840 (a variant of this sequence, when the order does matter for the equation x^2+y^2 = n, i.e. if the equation x^2+y^2 = n has got two solutions (x, y), (y, x) then they will be counted separately).
%Y Cf. A216672, A216674, A217834, A217840, A217956.
%K nonn
%O 1,2
%A _V. Raman_, Sep 13 2012
%E Ambiguity in name corrected by _V. Raman_, Oct 16 2012
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