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%I #7 Sep 08 2022 08:45:44
%S 53,73,125,193,365,697,1105,2117,4057,6437,12337,23645,37517,71905,
%T 137813,218665,419093,803233,1274473,2442653,4681585,7428173,14236825,
%U 27286277,43294565,82978297,159036077,252339217,483632957,926930185
%N Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x.
%C (-28, a(1)) and (A129289(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 1.
%H G. C. Greubel, <a href="/A160041/b160041.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=53, a(2)=73, a(3)=125, a(4)=193, a(5)=365, a(6)=697.
%F G.f.: (1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1-6*x^3+x^6).
%F a(3*k-1) = 73*A001653(k) for k >= 1.
%e (-28, a(1)) = (-28, 53) is a solution: (-28)^2+(-28+73)^2 = 784+2025 = 2809 = 53^2.
%e (A129289(1), a(2)) = (0, 73) is a solution: 0^2+(0+73)^2 = 5329 = 73^2.
%e (A129289(3), a(4)) = (95, 193) is a solution: 95^2+(95+73)^2 = 9025+28224 = 37249 = 193^2.
%t LinearRecurrence[{0,0,6,0,0,-1}, {53,73,125,193,365,697}, 50] (* _G. C. Greubel_, Apr 21 2018 *)
%o (PARI) {forstep(n=-28, 10000000, [3, 1], if(issquare(2*n^2+146*n+5329, &k), print1(k, ",")))}
%o (PARI) x='x+O('x^30); Vec((1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1 -6*x^3+x^6)) \\ _G. C. Greubel_, Apr 21 2018
%o (Magma) I:=[53,73,125,193,365,697]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, Apr 21 2018
%Y Cf. A129289, A001653, 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 nonn
%O 1,1
%A _Klaus Brockhaus_, May 04 2009
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