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A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

%I A130608 #13 Feb 15 2020 10:52:27
%S A130608 0,28,385,501,645,2668,3340,4176,15957,19873,24745,93408,116232,
%T A130608 144628,544825,677853,843357,3175876,3951220,4915848,18510765,
%U A130608 23029801,28652065,107889048,134227920,166996876,628823857,782338053,973329525,3665054428,4559800732
%N A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.
%C A130608 Also values x of Pythagorean triples (x, x+167, y).
%C A130608 Corresponding values y of solutions (x, y) are in A159777.
%C A130608 For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
%C A130608 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A130608 lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {1, 2}.
%C A130608 lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 0.
%H A130608 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A130608 a(n) = 6*a(n-3)-a(n-6)+334 for n > 6; a(1)=0, a(2)=28, a(3)=385, a(4)=501, a(5)=645, a(6)=2668.
%F A130608 G.f.: x*(28+357*x+116*x^2-24*x^3-119*x^4-24*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A130608 a(3*k+1) = 167*A001652(k) for k >= 0.
%t A130608 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,28,385,501,645,2668,3340},80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)
%o A130608 (PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+334*n+27889), print1(n, ",")))}
%Y A130608 Cf. A159777, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).
%K A130608 nonn,easy
%O A130608 1,2
%A A130608 _Mohamed Bouhamida_, Jun 17 2007
%E A130608 Edited and two terms added by _Klaus Brockhaus_, Apr 30 2009