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A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

%I A123654 #14 Sep 08 2022 08:45:28
%S A123654 0,264,1491,2427,3811,10764,16180,24220,64711,96271,143127,379120,
%T A123654 563064,836160,2211627,3283731,4875451,12892260,19140940,28418164,
%U A123654 75143551,111563527,165635151,437970664,650241840,965394360,2552682051
%N A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
%C A123654 Also values x of Pythagorean triples (x, x+809, y).
%C A123654 Corresponding values y of solutions (x, y) are in A160203.
%C A123654 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A123654 lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {1, 2}.
%C A123654 lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 0.
%H A123654 G. C. Greubel, <a href="/A123654/b123654.txt">Table of n, a(n) for n = 1..2500</a>
%H A123654 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A123654 a(n) = 6*a(n-3)-a(n-6)+1618 for n > 6; a(1)=0, a(2)=264, a(3)=1491, a(4)=2427, a(5)=3811, a(6)=10764.
%F A123654 G.f.: x*(264+1227*x+936*x^2-200*x^3-409*x^4-200*x^5) / ((1-x)*(1-6*x^3 +x^6)).
%F A123654 a(3*k+1) = 809*A001652(k) for k >= 0.
%t A123654 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,264,1491,2427,3811,10764,16180}, 50] (* _G. C. Greubel_, May 04 2018 *)
%o A123654 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481), print1(n, ",")))}
%o A123654 (PARI) x='x+O('x^30); concat([0], Vec(x*(264+1227*x+936*x^2-200*x^3 -409*x^4 -200*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ _G. C. Greubel_, May 04 2018
%o A123654 (Magma) I:=[0,264,1491,2427,3811,10764,16180]; [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 04 2018
%Y A123654 Cf. A160203, A001652, A115135, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).
%K A123654 nonn,easy
%O A123654 1,2
%A A123654 _Mohamed Bouhamida_, Jun 03 2007
%E A123654 Edited and two terms added by _Klaus Brockhaus_, May 18 2009