cp's OEIS Frontend

A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

%I A130005 #12 Feb 15 2020 10:52:27
%S A130005 0,35,1568,1731,1908,10595,11540,12567,63156,68663,74648,369495,
%T A130005 401592,436475,2154968,2342043,2545356,12561467,13651820,14836815,
%U A130005 73214988,79570031,86476688,426729615,463769520,504024467,2487163856,2703048243
%N A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.
%C A130005 Also values x of Pythagorean triples (x, x+577, y).
%C A130005 Corresponding values y of solutions (x, y) are in A159626.
%C A130005 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 A130005 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A130005 lim_{n -> infinity} a(n)/a(n-1) = (579+34*sqrt(2))/577 for n mod 3 = {1, 2}.
%C A130005 lim_{n -> infinity} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 0.
%H A130005 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A130005 a(n) = 6*a(n-3)-a(n-6)+1154 for n > 6; a(1)=0, a(2)=35, a(3)=1568, a(4)=1731, a(5)=1908, a(6)=10595.
%F A130005 G.f.: x*(35+1533*x+163*x^2-33*x^3-511*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A130005 a(3*k+1) = 577*A001652(k) for k >= 0.
%t A130005 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,35,1568,1731,1908,10595,11540},30] (* _Harvey P. Dale_, May 27 2018 *)
%o A130005 (PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+1154*n+332929), print1(n, ",")))}
%Y A130005 Cf. A159626, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).
%K A130005 nonn,easy
%O A130005 1,2
%A A130005 _Mohamed Bouhamida_, Jun 15 2007
%E A130005 Edited and two terms added by _Klaus Brockhaus_, Apr 21 2009