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A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

%I A129857 #18 Sep 08 2022 08:45:30
%S A129857 0,235,1696,2571,3796,12075,17140,24255,72468,101983,143448,424447,
%T A129857 596472,838147,2475928,3478563,4887148,14432835,20276620,28486455,
%U A129857 84122796,118182871,166033296,490305655,688822320,967715035,2857712848
%N A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.
%C A129857 Also values x of Pythagorean triples (x, x+857, y).
%C A129857 Corresponding values y of solutions (x, y) are in A160206.
%C A129857 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129857 lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.
%C A129857 lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 0.
%H A129857 G. C. Greubel, <a href="/A129857/b129857.txt">Table of n, a(n) for n = 1..1000</a>
%H A129857 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129857 a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.
%F A129857 G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A129857 a(3*k+1) = 857*A001652(k) for k >= 0.
%t A129857 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* _Harvey P. Dale_, Apr 24 2011 *)
%o A129857 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}
%o A129857 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // _G. C. Greubel_, May 03 2018
%Y A129857 Cf. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).
%K A129857 nonn,easy
%O A129857 1,2
%A A129857 _Mohamed Bouhamida_, Jun 03 2007
%E A129857 Edited and two terms added by _Klaus Brockhaus_, May 18 2009