A216672 Total number of solutions to the equation x^2 + k*y^2 = n with x > 0, y > 0, k > 0. (Order does not matter for the equation x^2 + y^2 = n.)

0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 3, 3, 4, 6, 5, 5, 6, 6, 5, 4, 6, 7, 5, 6, 8, 8, 5, 6, 8, 9, 7, 5, 9, 10, 6, 6, 10, 11, 6, 8, 9, 11, 7, 6, 10, 11, 8, 8, 14, 11, 10, 8, 10, 13, 9, 8, 10, 14, 7, 9, 12, 14, 9, 10, 14, 12, 10, 8, 15, 17, 9, 9, 16, 12, 8, 11

Offset: 1

Author

V. Raman, Sep 13 2012

Keywords

• nonn

Comments

If the equation x^2 + y^2 = n has 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 A216503 since this sequence gives the total number of solutions to the equation x^2 + k*y^2 = n, whereas the sequence A216503 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.

So for this sequence a(33) = 9.

On the other hand, for the sequence A216503, there exist only 7 different values of k for which a solution to the equation mentioned above exists.

So A216503(33) = 7.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A216503, A216504, A216505.

Cf. A217834 (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. A216673, A216674, A217834, A217840, A217956.

Programs

Mathematica

nn = 100; t = Table[0, {nn}]; Do[n = x^2 + k*y^2; If[n <= nn && (k > 1 || k == 1 && x <= y), t[[n]]++], {x, Sqrt[nn]}, {y, Sqrt[nn]}, {k, nn}] (* T. D. Noe, Sep 20 2012 *)

PARI

for(n=1, 100, sol=0; for(k=1, n, for(x=1, 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++))); print1(sol", ")) /* V. Raman, Oct 16 2012 */

Extensions

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