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

29, 45232349, 189482250299273866821980904657123150749

Offset

0,1

Comments

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

The product expansion in question is sqrt(2) = product {n = 0..inf} (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

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.

Mathematica

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

Crossrefs

Cf. A000129, A219011, A219013, A219014.

Sequence in context: A303734 A087528 A307622 * A144839 A124992 A296522

Adjacent sequences: A219012 A219013 A219014 * A219016 A219017 A219018

Keyword

nonn,easy,bref

Author

Peter Bala, Nov 09 2012

Status

approved