cp's OEIS Frontend

A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

%I A129999 #13 Feb 15 2020 10:52:27
%S A129999 0,27,888,1011,1148,6027,6740,7535,35948,40103,44736,210335,234552,
%T A129999 261555,1226736,1367883,1525268,7150755,7973420,8890727,41678468,
%U A129999 46473311,51819768,242920727,270867120,302028555,1415846568,1578730083
%N A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.
%C A129999 Also values x of Pythagorean triples (x, x+337, y).
%C A129999 Corresponding values y of solutions (x, y) are in A159574.
%C A129999 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 A129999 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129999 lim_{n -> infinity} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.
%C A129999 lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.
%H A129999 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129999 a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.
%F A129999 G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A129999 a(3*k+1) = 337*A001652(k) for k >= 0.
%F A129999 a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - _Harvey P. Dale_, Feb 26 2015
%t A129999 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* _Harvey P. Dale_, Feb 26 2015 *)
%o A129999 (PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}
%Y A129999 Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).
%K A129999 nonn,easy
%O A129999 1,2
%A A129999 _Mohamed Bouhamida_, Jun 15 2007
%E A129999 Edited and two terms added by _Klaus Brockhaus_, Apr 16 2009