A217834 Total number of solutions to the equation x^2+k*y^2 = n with x > 0, y > 0, k > 0. (Order matters for the equation x^2+y^2 = n).

0, 1, 1, 1, 3, 2, 2, 3, 3, 4, 3, 4, 6, 3, 3, 4, 7, 5, 5, 7, 6, 5, 4, 6, 8, 6, 6, 8, 9, 5, 6, 8, 9, 8, 5, 9, 11, 6, 6, 11, 12, 6, 8, 9, 12, 7, 6, 10, 11, 9, 8, 15, 12, 10, 8, 10, 13, 10, 8, 10, 15, 7, 9, 12, 16, 9, 10, 15, 12, 10, 8, 15, 18, 10, 9, 16, 12, 8, 11, 15, 17, 12, 9, 16, 19, 10, 9, 16, 18, 13, 12, 13, 14, 11, 9, 15, 21, 10, 14, 20

Offset

1,5

Comments

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

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

a(n) is the same as A216672(n) when n is not the sum of two positive squares.

But when n is the sum of two positive squares, the ordered pairs for the equation x^2+y^2 = n count.

For example,

10 = 3^2 + 1^2.

10 = 1^2 + 3^2.

10 = 2^2 + 6*1^2.

10 = 1^2 + 9*1^2.

So a(10) = 4. On the other hand, for the sequence A216672, the ordered pair 3^2+1^2 and 1^2+3^2 will be counted as the same, and so A216672(10) = 3.

Links

Table of n, a(n) for n=1..100.

Prog

(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), sol++))); print1(sol", ")) /* V. Raman, Oct 16 2012 */

Crossrefs

Cf. A216503, A216504, A216505.

Cf. A216672 (a variant of this sequence, when the order does not 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 as the same).

Cf. A216672, A216673, A216674, A217840, A217956.

Sequence in context: A115039 A032536 A115061 * A227877 A225867 A046822

Adjacent sequences: A217831 A217832 A217833 * A217835 A217836 A217837

Keyword

nonn

Author

V. Raman, Oct 16 2012

Status

approved