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A129992 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.

%I A129992 #16 Sep 08 2022 08:45:30
%S A129992 0,17,308,381,468,2117,2540,3045,12648,15113,18056,74025,88392,105545,
%T A129992 431756,515493,615468,2516765,3004820,3587517,14669088,17513681,
%U A129992 20909888,85498017,102077520,121872065,498319268,594951693,710322756,2904417845,3467632892
%N A129992 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.
%C A129992 Also values x of Pythagorean triples (x, x+127, y).
%C A129992 Corresponding values y of solutions (x, y) are in A159466.
%C A129992 For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
%C A129992 Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129992 Lim_{n -> infinity} a(n)/a(n-1) = (129+16*sqrt(2))/127 for n mod 3 = {1, 2}.
%C A129992 lim_{n -> infinity} a(n)/a(n-1) = (34947+21922*sqrt(2))/127^2 for n mod 3 = 0.
%H A129992 G. C. Greubel, <a href="/A129992/b129992.txt">Table of n, a(n) for n = 1..1000</a>
%H A129992 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 6, -6, 0, -1, 1).
%F A129992 a(n) = 6*a(n-3) - a(n-6) + 254 for n > 6; a(1)=0, a(2)=17, a(3)=308, a(4)=381, a(5)=468, a(6)=2117.
%F A129992 G.f.: x*(17+291*x+73*x^2-15*x^3-97*x^4-15*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A129992 a(3*k+1) = 127*A001652(k) for k >= 0.
%t A129992 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,17,308,381,468,2117,2540},80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)
%o A129992 (PARI) {forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129), print1(n, ",")))};
%o A129992 (Magma) I:=[0,17,308,381,468,2117,2540]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // _G. C. Greubel_, Mar 31 2018
%Y A129992 Cf. A159466, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).
%K A129992 nonn
%O A129992 1,2
%A A129992 _Mohamed Bouhamida_, Jun 14 2007
%E A129992 Edited and two more terms added by _Klaus Brockhaus_, Apr 13 2009