A219013 - OEIS

%I #8 Nov 03 2013 08:24:24

%S 11,523451,39571031999225940638473470251

%N Denominators in a product expansion for sqrt(3).

%C The product expansion in question is sqrt(3) = product {n = 0..inf} (1 + 2*A219012(n)/A219013(n)) = (1 + 2*4/11)*(1 + 2*724/523451)*(1 + 2*198924689265124/39571031999225940638473470251)*....

%F Let alpha = 1/2*(sqrt(2) + sqrt(6)) and put f(n) = 1/sqrt(6)*{alpha^n - (-1/alpha)^n}. Then a(n) = f(5^(n+1))/f(5^n).

%F a(n) = A219012(n)^2 - A219012(n) - 1.

%F Recurrence equation: a(n+1) = 5/2*(a(n)^4 - a(n)^2)*sqrt(4*a(n) + 5) + a(n)^5 + 15/2*a(n)^4 - 25/2*a(n)^2 + 5 with initial condition a(0) = 11.

%Y Cf. A219011, A219012, A219015.

%K nonn,easy,bref

%O 0,1

%A _Peter Bala_, Nov 09 2012