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%I #2 Mar 30 2012 17:28:01
%S 85,113,173,337,565,953,1937,3277,5545,11285,19097,32317,65773,111305,
%T 188357,383353,648733,1097825,2234345,3781093,6398593,13022717,
%U 22037825,37293733,75901957,128445857,217363805,442389025,748637317,1266889097
%N Positive numbers y such that y^2 is of the form x^2+(x+113)^2 with integer x.
%C (-36, a(1)) and (A161478(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+113)^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) = (129+44*sqrt(2))/113 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 1.
%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=85, a(2)=113, a(3)=173, a(4)=337, a(5)=565, a(6)=953.
%F G.f.: (1-x)*(85+198*x+371*x^2+198*x^3+85*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 113*A001653(k) for k >= 1.
%e (-36, a(1)) = (-36, 85) is a solution: (-36)^2+(-36+113)^2 = 1296+5929 = 7225 = 85^2.
%e (A161478(1), a(2)) = (0, 113) is a solution: 0^2+(0+113)^2 = 12769 = 113^2.
%e (A161478(3), a(4)) = (175, 337) is a solution: 175^2+(175+113)^2 = 30625+82944 = 113569 = 337^2.
%o (PARI) {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+226*n+12769, &k), print1(k, ",")))}
%Y Cf. A161478, A001653, 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 1,1
%A _Klaus Brockhaus_, Jun 13 2009
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