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A239434 Number of nonnegative integer solutions to the equation x^2 - 25*y^2 = n. 2

%I #9 Oct 30 2014 07:31:02

%S 1,0,0,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,0,1,0,0,0,

%T 0,1,0,0,1,0,1,0,0,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1,0,0,2,0,0,0,0,

%U 1,0,1,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0

%N Number of nonnegative integer solutions to the equation x^2 - 25*y^2 = n.

%C For (x, y) to be a solution to the more general equation x^2 - d^2*y^2 = n, it can be shown that n-f^2 must be divisible by 2*f*d, where f is a divisor of n not exceeding sqrt(n). Then y = (n-f^2)/(2*f*d) and x = d*y+f.

%H Colin Barker, <a href="/A239434/b239434.txt">Table of n, a(n) for n = 1..1000</a>

%e a(64)=2 because x^2 - 25*y^2 = 64 has two solutions, (X,Y) = (8,0) and (17,3).

%o (PARI) a(n) = sumdiv(n, f, f^2<=n && (n-f^2)%(10*f)==0)

%Y Cf. A034178, A230240, A230263, A230264, A239435.

%K nonn

%O 1,64

%A _Colin Barker_, Mar 18 2014

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