|
%I #15 Feb 15 2020 10:52:27
%S 0,45,2688,2901,3128,18105,19340,20657,107876,115073,122748,631085,
%T 673032,717765,3680568,3925053,4185776,21454257,22879220,24398825,
%U 125046908,133352201,142209108,728829125,777235920,828857757,4247929776
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+967, y).
%C Corresponding values y of solutions (x, y) are in A159701.
%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) = (969+44**sqrt(2))/967 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 0.
%H Harvey P. Dale, <a href="/A130017/b130017.txt">Table of n, a(n) for n = 1..1000</a>
%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)+1934 for n > 6; a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105.
%F G.f.: x*(45+2643*x+213*x^2-43*x^3-881*x^4-43*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 967*A001652(k) for k >= 0.
%F a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105, a(7)=19340, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - _Harvey P. Dale_, Nov 03 2013
%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,45,2688,2901,3128,18105,19340},40] (* _Harvey P. Dale_, Nov 03 2013 *)
%o (PARI) {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089), print1(n, ",")))}
%Y Cf. A159701, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44**sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, Jun 15 2007
%E Edited and two terms added by _Klaus Brockhaus_, Apr 21 2009
|
