cp's OEIS Frontend

A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

%I A129991 #17 Feb 15 2020 10:52:27
%S A129991 0,23,620,723,840,4223,4820,5499,25200,28679,32636,147459,167736,
%T A129991 190799,860036,978219,1112640,5013239,5702060,6485523,29219880,
%U A129991 33234623,37800980,170306523,193706160,220320839,992619740,1129002819,1284124536,5785412399,6580311236
%N A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.
%C A129991 Also values x of Pythagorean triples (x, x+241, y).
%C A129991 Corresponding values y of solutions (x, y) are in A159565.
%C A129991 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 A129991 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A129991 lim_{n -> infinity} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {1, 2}.
%C A129991 lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 0.
%H A129991 Vincenzo Librandi, <a href="/A129991/b129991.txt">Table of n, a(n) for n = 1..1000</a>
%H A129991 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A129991 a(n) = 6*a(n-3)-a(n-6)+482 for n > 6; a(1)=0, a(2)=23, a(3)=620, a(4)=723, a(5)=840, a(6)=4223.
%F A129991 G.f.: x*(23+597*x+103*x^2-21*x^3-199*x^4-21*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A129991 a(3*k+1) = 241*A001652(k) for k >= 0.
%t A129991 LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 23, 620, 723, 840, 4223, 4820}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2012 *)
%o A129991 (PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+482*n+58081), print1(n, ",")))}
%Y A129991 Cf. A159565, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).
%K A129991 nonn,easy
%O A129991 1,2
%A A129991 _Mohamed Bouhamida_, Jun 14 2007
%E A129991 Edited and two terms added by _Klaus Brockhaus_, Apr 16 2009