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A129725 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2. 4

%I #14 Feb 15 2020 10:52:27

%S 0,100,1159,1563,2079,8080,10420,13416,48363,61999,79459,283140,

%T 362616,464380,1651519,2114739,2707863,9627016,12326860,15783840,

%U 56111619,71847463,91996219,327043740,418758960,536194516,1906151863,2440707339,3125171919,11109868480

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+521, y).

%C Corresponding values y of solutions (x, y) are in A160583.

%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 0.

%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)+1042 for n > 6; a(1)=0, a(2)=100, a(3)=1159, a(4)=1563, a(5)=2079, a(6)=8080.

%F G.f.: x*(100+1059*x+404*x^2-84*x^3-353*x^4-84*x^5) / ((1-x)*(1-6*x^3+x^6)).

%F a(3*k+1) = 521*A001652(k) for k >= 0.

%t LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 100, 1159, 1563, 2079, 8080, 10420}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)

%o (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441), print1(n, ",")))}

%Y Cf. A160583, A001652, A129642, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

%K nonn,easy

%O 1,2

%A _Mohamed Bouhamida_, Jun 02 2007

%E Edited and two terms added by _Klaus Brockhaus_, Jun 08 2009

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