|
|
A219015
|
|
Denominators in a product expansion for sqrt(2).
|
|
4
|
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,bref
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|