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%I #2 Mar 30 2012 17:28:01
%S 109,151,265,389,755,1481,2225,4379,8621,12961,25519,50245,75541,
%T 148735,292849,440285,866891,1706849,2566169,5052611,9948245,14956729,
%U 29448775,57982621,87174205,171640039,337947481,508088501,1000391459,1969702265
%N Positive numbers y such that y^2 is of the form x^2+(x+151)^2 with integer x.
%C (-60, a(1)) and (A161482(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+151)^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) = (187+78*sqrt(2))/151 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (24723+6758*sqrt(2))/151^2 for n mod 3 = 1.
%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=109, a(2)=151, a(3)=265, a(4)=389, a(5)=755, a(6)=1481.
%F G.f.: (1-x)*(109+260*x+525*x^2+260*x^3+109*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 151*A001653(k) for k >= 1.
%e (-60, a(1)) = (-60, 109) is a solution: (-60)^2+(-60+151)^2 = 3600+8281 = 11881 = 109^2.
%e (A161482(1), a(2)) = (0, 151) is a solution: 0^2+(0+151)^2 = 22801 = 151^2.
%e (A161482(3), a(4)) = (189, 389) is a solution: 189^2+(189+151)^2 = 35721+115600 = 151321 = 389^2.
%o (PARI) {forstep(n=-60, 100000000, [1, 3], if(issquare(2*n^2+302*n+22801, &k), print1(k, ",")))}
%Y Cf. A161482, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161484 (decimal expansion of (187+78*sqrt(2))/151), A161485 (decimal expansion of (24723+6758*sqrt(2))/151^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Jun 13 2009
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