cp's OEIS Frontend

A216674 Total number of solutions to the equation x^2+k*y^2 = n with x > 0, y > 0, k >= 0, or 0 if infinite. (Order does not matter for the equation x^2+y^2 = n).

%I A216674 #29 Feb 22 2018 18:00:04
%S A216674 0,1,1,0,2,2,2,3,0,3,3,4,5,3,3,0,6,5,5,6,6,5,4,6,0,5,6,8,8,5,6,8,9,7,
%T A216674 5,0,10,6,6,10,11,6,8,9,11,7,6,10,0,8,8,14,11,10,8,10,13,9,8,10,14,7,
%U A216674 9,0,14,9,10,14,12,10,8,15,17,9,9,16,12,8,11,14,0
%N A216674 Total number of solutions to the equation x^2+k*y^2 = n with x > 0, y > 0, k >= 0, or 0 if infinite. (Order does not matter for the equation x^2+y^2 = n).
%C A216674 If the equation x^2+y^2 = n has got two solutions (x, y), (y, x) then they will be counted only once.
%C A216674 No solutions can exist for the values of k >= n.
%C A216674 This sequence differs from A216505 in the fact that this sequence gives the total number of solutions to the equation x^2+k*y^2 = n, whereas the sequence A216505 gives the number of distinct values of k for which a solution to the equation x^2+k*y^2 = n can exist.
%C A216674 Some values of k can clearly have more than one solution.
%C A216674 For example, x^2+k*y^2 = 33 is satisfiable for
%C A216674 33 = 1^2+2*4^2.
%C A216674 33 = 5^2+2*2^2.
%C A216674 33 = 3^2+6*2^2.
%C A216674 33 = 1^2+8*2^2.
%C A216674 33 = 5^2+8*1^2.
%C A216674 33 = 4^2+17*1^2.
%C A216674 33 = 3^2+24*1^2.
%C A216674 33 = 2^2+29*1^2.
%C A216674 33 = 1^2+32*1^2.
%C A216674 So for this sequence a(33) = 9.
%C A216674 On the other hand, for the sequence A216505, there exist only 7 different values of k for which a solution to the equation mentioned above exists.
%C A216674 So A216505(33) = 7.
%H A216674 Charles R Greathouse IV, <a href="/A216674/b216674.txt">Table of n, a(n) for n = 1..10000</a>
%o A216674 (PARI) a(n)=if(issquare(n),return(0));sum(y=ceil(sqrt(n/2-1/4)), sqrtint(n-1),issquare(n-y^2))+sum(k=2,n-1,sum(y=1,sqrtint((n-1)\k), issquare(n-k*y^2))) \\ _Charles R Greathouse IV_, Sep 14 2012
%o A216674 (PARI) for(n=1, 100, sol=0; for(k=0, 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++))); if(issquare(n),print1(0", "),print1(sol", "))) /* _V. Raman_, Oct 16 2012 */
%Y A216674 Cf. A216503, A216504, A216505.
%Y A216674 Cf. A217956 (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 A216674 Cf. A216672, A216673, A217834, A217840, A217956.
%K A216674 nonn
%O A216674 1,5
%A A216674 _V. Raman_, Sep 13 2012
%E A216674 Ambiguity in name corrected by _V. Raman_, Oct 16 2012