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A160212 Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x. 3

%I

%S 845,953,1093,3977,4765,5713,23017,27637,33185,134125,161057,193397,

%T 781733,938705,1127197,4556273,5471173,6569785,26555905,31888333,

%U 38291513,154779157,185858825,223179293,902119037,1083264617,1300784245

%N Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x.

%C (-116, a(1)) and (A129975(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

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

%C lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {0, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 1.

%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=845, a(2)=953, a(3)=1093, a(4)=3977, a(5)=4765, a(6)=5713.

%F G.f.: (1-x)*(845+1798*x+2891*x^2+1798*x^3+845*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 953*A001653(k) for k >= 1.

%e (-116, a(1)) = (-116, 845) is a solution: (-116)^2+(-116+953)^2 = 13456+700569 = 714025 = 845^2.

%e (A129975(1), a(2)) = (0, 953) is a solution: 0^2+(0+953)^2 = 908209 = 953^2.

%e (A129975(3), a(4)) = (2295, 3977) is a solution: 2295^2+(2295+953)^2 = 5267025+10549504 = 15816529 = 3977^2.

%o (PARI) {forstep(n=-116, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209, &k), print1(k, ",")))}

%Y Cf. A129975, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

%K nonn

%O 1,1

%A _Klaus Brockhaus_, May 18 2009

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