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A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

%I A129289 #15 Sep 08 2022 08:45:30
%S A129289 0,44,95,219,455,744,1460,2832,4515,8687,16683,26492,50808,97412,
%T A129289 154583,296307,567935,901152,1727180,3310344,5252475,10066919,
%U A129289 19294275,30613844,58674480,112455452,178430735,341980107,655438583,1039970712,1993206308,3820176192
%N A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.
%C A129289 Also values x of Pythagorean triples (x, x+73, y).
%C A129289 Corresponding values y of solutions (x, y) are in A160041.
%C A129289 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129289 lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {1, 2}.
%C A129289 lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 0.
%H A129289 G. C. Greubel, <a href="/A129289/b129289.txt">Table of n, a(n) for n = 1..1000</a>
%H A129289 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129289 a(n) = 6*a(n-3) -a(n-6) +146 for n > 6; a(1)=0, a(2)=44, a(3)=95, a(4)=219, a(5)=455, a(6)=744.
%F A129289 G.f.: x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A129289 a(3*k+1) = 73*A001652(k) for k >= 0.
%t A129289 Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+73)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,44,95,219,455,744,1460},70] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%o A129289 (PARI) {forstep(n=0, 100000000, [3 ,1], if(issquare(2*n^2+146*n+5329), print1(n, ",")))}
%o A129289 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)))); // _G. C. Greubel_, May 07 2018
%Y A129289 Cf. A160041, A129288, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).
%K A129289 nonn
%O A129289 1,2
%A A129289 _Mohamed Bouhamida_, May 26 2007
%E A129289 Edited and two terms added by _Klaus Brockhaus_, May 04 2009