|
%I #15 Sep 08 2022 08:45:30
%S 0,60,2241,2517,2821,15180,16780,18544,90517,99841,110121,529600,
%T 583944,643860,3088761,3405501,3754717,18004644,19850740,21886120,
%U 104940781,115700617,127563681,611641720,674354640,743497644,3564911217
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+839, y).
%C Corresponding values y of solutions (x, y) are in A159896.
%C 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 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 0.
%H G. C. Greubel, <a href="/A130647/b130647.txt">Table of n, a(n) for n = 1..3895</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) +1678 for n > 6; a(1)=0, a(2)=60, a(3)=2241, a(4)=2517, a(5)=2821, a(6)=15180.
%F G.f.: x*(60+2181*x+276*x^2-56*x^3-727*x^4-56*x^5)/((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 839*A001652(k) for k >= 0.
%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,60,2241,2517,2821,15180,16780},30] (* _Harvey P. Dale_, Jun 19 2014 *)
%o (PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ",")))}
%o (Magma) I:=[0,60,2241,2517,2821,15180,16780]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n=7): n in [1..30]]; // _G. C. Greubel_, May 17 2018
%Y Cf. A159896, A028871, A118337, A130645, A130646, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, Jun 20 2007
%E Edited and two terms added by _Klaus Brockhaus_, Apr 30 2009
|
