A219015 Denominators in a product expansion for sqrt(2).

29, 45232349, 189482250299273866821980904657123150749

Offset: 0

Author

Peter Bala, Nov 09 2012

Keywords

• nonn
• easy
• bref

Comments

a(3) has 192 digits and a(4) has 957 digits.

The product expansion in question is sqrt(2) = Product_{n = 0..infinity} (1 + 2*A219014(n)/A219015(n)) = (1 + 2*6/29)*(1 + 2*6726/45232349)*....

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4

Crossrefs

Cf. A000129, A219011, A219013, A219014.

Programs

Mathematica

Table[Fibonacci[5^(n+1),2]/Fibonacci[5^n,2], {n,0,5}] (* G. C. Greubel, Feb 02 2018 *)

Formula

a(n) = Pell(5^(n+1))/Pell(5^n), where Pell(n) = A000129(n).

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) = 29.