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A111258 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.

%I A111258 #15 Sep 08 2022 08:45:21
%S A111258 0,539,560,1803,4740,4859,12020,29103,29796,71519,171080,175119,
%T A111258 418296,998579,1022120,2439459,5821596,5958803,14219660,33932199,
%U A111258 34731900,82879703,197772800,202433799,483059760,1152705803,1179872096
%N A111258 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.
%C A111258 Also values x of Pythagorean triples (x, x+601, y).
%C A111258 Corresponding values y of solutions (x, y) are in A160098.
%C A111258 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A111258 lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {1, 2}.
%C A111258 lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 0.
%H A111258 G. C. Greubel, <a href="/A111258/b111258.txt">Table of n, a(n) for n = 1..1000</a>
%H A111258 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A111258 a(n) = 6*a(n-3) - a(n-6) + 1202 for n > 6; a(1)=0, a(2)=539, a(3)=560, a(4)=1803, a(5)=4740, a(6)=4859.
%F A111258 G.f.: x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)).
%F A111258 a(3*k+1) = 601*A001652(k) for k >= 0.
%t A111258 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,539,560,1803,4740,4859,12020}, 50] (* _G. C. Greubel_, Apr 22 2018 *)
%o A111258 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201), print1(n, ",")))}
%o A111258 (PARI) x='x+O('x^30); concat([0], Vec(x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)))) \\ _G. C. Greubel_, Apr 22 2018
%o A111258 (Magma) I:=[0,539,560,1803,4740,4859,12020]; [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_, Apr 22 2018
%Y A111258 Cf. A160098, A001652, A101152, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).
%K A111258 nonn,easy
%O A111258 1,2
%A A111258 _Mohamed Bouhamida_, Jun 03 2007
%E A111258 Edited and one term added by _Klaus Brockhaus_, May 18 2009