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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.
5

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

%S 0,37,1768,1941,2128,11937,12940,14025,71148,76993,83316,416245,

%T 450312,487165,2427616,2626173,2840968,14150745,15308020,16559937,

%U 82478148,89223241,96519948,480719437,520032720,562561045,2801839768,3030974373

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+647, y).

%C Corresponding values y of solutions (x, y) are in A159641.

%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) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 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)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.

%F G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).

%F a(3*k+1) = 647*A001652(k) for k >= 0.

%o (PARI) {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), print1(n, ",")))}

%Y Cf. A159641, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

%K nonn,easy

%O 1,2

%A _Mohamed Bouhamida_, Jun 15 2007

%E Edited and two terms added by _Klaus Brockhaus_, Apr 21 2009