A216673 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).

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, 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, 15, 8, 11, 0, 15, 10, 11, 16, 13, 11, 9, 19, 18, 10, 11, 18, 13

Offset: 1

Author

V. Raman, Sep 13 2012

Keywords

• nonn

Comments

If the equation x^2+y^2 = n has got two solutions (x, y), (y, x) then they will be counted only once.

No solutions can exist for the values of k > n.

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.

Some values of k can clearly have more than one solution.

For example, x^2+k*y^2 = 33 is satisfiable for

33 = 1^2+2*4^2.

33 = 5^2+2*2^2.

33 = 3^2+6*2^2.

33 = 1^2+8*2^2.

33 = 5^2+8*1^2.

33 = 4^2+17*1^2.

33 = 3^2+24*1^2.

33 = 2^2+29*1^2.

33 = 1^2+32*1^2.

33 = 0^2+33*1^2.

So for this sequence a(33) = 10.

On the other hand, for A216504, there exist only 7 different values of k for which a solution to the equation mentioned above exists.

So A216504(33) = 8.

Crossrefs

Cf. A216503, A216504, A216505.

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).

Cf. A216672, A216674, A217834, A217840, A217956.

Programs

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 */

Extensions

Ambiguity in name corrected by V. Raman, Oct 16 2012