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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.
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%I #8 Jul 31 2015 22:13:52

%S 0,52,175,339,615,1312,2260,3864,7923,13447,22795,46452,78648,133132,

%T 271015,458667,776223,1579864,2673580,4524432,9208395,15583039,

%U 26370595,53670732,90824880,153699364,312816223,529366467,895825815,1823226832,3085374148

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

%C Corresponding values y of solutions (x, y) are in A161479.

%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 0.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 6, -6, 0, -1, 1).

%F a(n) = 6*a(n-3)-a(n-6)+226 for n > 6; a(1)=0, a(2)=52, a(3)=175, a(4)=339, a(5)=615, a(6)=1312.

%F G.f.: x*(52+123*x+164*x^2-36*x^3-41*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).

%F a(3*k+1) = 113*A001652(k) for k >= 0.

%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,52,175,339,615,1312,2260},72] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)

%o (PARI) {forstep(n=0, 100000000, [3, 1], if(issquare(2*n^2+226*n+12769), print1(n, ",")))}

%Y Cf. A161479, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

%K nonn

%O 1,2

%A _Klaus Brockhaus_, Jun 13 2009