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%I #19 Sep 08 2022 08:45:30
%S 0,31,1204,1347,1504,8151,8980,9891,48600,53431,58740,284347,312504,
%T 343447,1658380,1822491,2002840,9666831,10623340,11674491,56343504,
%U 61918447,68045004,328395091,360888240,396596431,1914027940,2103411891,2311534480,11155773447
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+449, y).
%C Corresponding values y of solutions (x, y) are in A159589.
%C 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 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 0.
%H G. C. Greubel, <a href="/A130004/b130004.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F a(n) = 6*a(n-3) -a(n-6) +898 for n > 6; a(1)=0, a(2)=31, a(3)=1204, a(4)=1347, a(5)=1504, a(6)=8151.
%F G.f.: x*(31+1173*x+143*x^2-29*x^3-391*x^4-29*x^5)/((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 449*A001652(k) for k >= 0.
%t LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2012 *)
%o (PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ",")))}
%o (PARI) x='x+O('x^30); concat([0], Vec(x*(31+1173*x+143*x^2-29*x^3-391*x^4 -29*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ _G. C. Greubel_, May 08 2018
%o (Magma) I:=[0, 31, 1204, 1347, 1504, 8151, 8980]; [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..30]]; // _G. C. Greubel_, May 08 2018
%Y Cf. A159589, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, Jun 15 2007
%E Edited and two terms added by _Klaus Brockhaus_, Apr 17 2009