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

5, 15005, 792070839820228500005

Offset

0,1

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A364691 A367943 A089989 * A297482 A185686 A083976

Adjacent sequences: A219008 A219009 A219010 * A219012 A219013 A219014

Keyword

nonn,easy

Author

Peter Bala, Nov 09 2012

Status

approved