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A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

%I A118675 #16 Sep 08 2022 08:45:25
%S A118675 0,16,85,141,225,616,940,1428,3705,5593,8437,21708,32712,49288,126637,
%T A118675 190773,287385,738208,1112020,1675116,4302705,6481441,9763405,
%U A118675 25078116,37776720,56905408,146166085,220178973,331669137,851918488,1283297212,1933109508
%N A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.
%C A118675 Also values x of Pythagorean triples (x, x+47, y).
%C A118675 Corresponding values y of solutions (x, y) are in A159750.
%C A118675 For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
%C A118675 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A118675 lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {1, 2}.
%C A118675 lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 0.
%H A118675 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A118675 a(n) = 6*a(n-3) -a(n-6) +94 for n > 6; a(1)=0, a(2)=16, a(3)=85, a(4)=141, a(5)=225, a(6)=616.
%F A118675 G.f.: x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A118675 a(3*k+1) = 47*A001652(k) for k >= 0.
%t A118675 Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+47)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,16,85,141,225,616,940},50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%o A118675 (PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+94+2209), print1(n, ",")))}
%o A118675 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3 +x^6)))); // _G. C. Greubel_, May 07 2018
%Y A118675 Cf. A159750, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).
%K A118675 nonn
%O A118675 1,2
%A A118675 _Mohamed Bouhamida_, May 19 2006
%E A118675 Edited by _Klaus Brockhaus_, Apr 30 2009