A219011 Denominators in a product expansion for sqrt(5).

5, 15005, 792070839820228500005

Offset: 0

Author

Peter Bala, Nov 09 2012

Keywords

• nonn
• easy

Comments

Apart from the initial term same as A145275.

a(3) has 105 digits and a(4) has 523 digits.

The product expansion in question is sqrt(5) = Product_{n >= 0} (1 + 2*A219010(n)/A219011(n)) = (1 + 6/5)*(1 + 246/15005)*(1 + 56287506246/792070839820228500005)*....

Links

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

Crossrefs

Cf. A000032, A000045, A145275, A219010, A219013, A219015.

Programs

Mathematica

a[n_] := LucasL[4*5^n] - LucasL[2*5^n] + 1; Array[a, 3, 0] (* Amiram Eldar, Jul 20 2025 *)

Maxima

A219011(n):=fib(5^(n+1))/fib(5^n)$
makelist(A219011(n),n,0,3);

Formula

a(n) = Fibonacci(5^(n+1))/Fibonacci(5^n).

a(n) = A219010(n)^2 - A219010(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) = 5.

a(n) = Lucas(4*5^n) - Lucas(2*5^n) + 1. - Ehren Metcalfe, Jul 29 2017