|
%I #2 Mar 30 2012 17:28:01
%S 641,809,1105,2741,4045,5989,15805,23461,34829,92089,136721,202985,
%T 536729,796865,1183081,3128285,4644469,6895501,18232981,27069949,
%U 40189925,106269601,157775225,234244049,619384625,919581401,1365274369
%N Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x.
%C (-200, a(1)) and (A123654(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 1.
%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=641, a(2)=809, a(3)=1105, a(4)=2741, a(5)=4045, a(6)=5989.
%F G.f.: (1-x)*(641+1450*x+2555*x^2+1450*x^3+641*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 809*A001653(k) for k >= 1.
%e (-200, a(1)) = (-200, 641) is a solution: (-200)^2+(-200+809)^2 = 40000+370881 = 410881 = 641^2.
%e (A123654(1), a(2)) = (0, 809) is a solution: 0^2+(0+809)^2 = 654481 = 809^2.
%e (A123654(3), a(4)) = (1491, 2741) is a solution: 1491^2+(1491+809)^2 = 2223081+5290000 = 7513081 = 2741^2.
%o (PARI) {forstep(n=-200, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481, &k), print1(k, ",")))}
%Y Cf. A123654, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, May 18 2009
|
