A219013 Denominators in a product expansion for sqrt(3).

11, 523451, 39571031999225940638473470251

Offset: 0

Author

Peter Bala, Nov 09 2012

Keywords

• nonn
• easy

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 0..4

Crossrefs

Cf. A188887, A219011, A219012, A219015.

Programs

Mathematica

RecurrenceTable[{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, a[0] == 11}, a, {n, 0, 3}] (* Amiram Eldar, Jul 20 2025 *)

Formula

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).

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

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.