

A120994


Numerators of rationals related to John Wallis' product formula for Pi/2 from his 'Arithmetica infinitorum' from 1659.


3



1, 16, 192, 4096, 16384, 262144, 1048576, 268435456, 3221225472, 17179869184, 68719476736, 13194139533312, 17592186044416, 281474976710656, 1125899906842624, 1152921504606846976, 4611686018427387904
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OFFSET

1,2


COMMENTS

The corresponding denominators are given in A120995.
The normalized sequence of rationals r(n):=(3/4)*W(n), with r(1)=1, converges to 3*Pi/8 = 1.178097245...
The product formula for Pi/2 of Wallis can be written like lim_{n to infinity} W(n) with the rationals W(n):=(((2*n)!!/(2*n1)!!)^2)/(2*n+1) with the double factorials (2*n)!! = A000165(n) and (2*n1)!! = A001147(n).


LINKS

Table of n, a(n) for n=1..17.
W. Lang: Rationals r(n) and limit.


FORMULA

a(n) = numerator((3/4)*W(n)), n>=1, with the rationals W(n) given above. An equivalent form is W(n) = (((4^n)/binomial(2*n,n))^2)/(2*n+1).


EXAMPLE

Rationals r(n)=((3/4)*W(n)): [1, 16/15, 192/175, 4096/3675,
16384/14553, 262144/231231, 1048576/920205, 268435456/234652275,...]


CROSSREFS

Sequence in context: A000767 A053539 A218176 * A016178 A081202 A196803
Adjacent sequences: A120991 A120992 A120993 * A120995 A120996 A120997


KEYWORD

nonn,easy,frac


AUTHOR

Wolfdieter Lang, Aug 01 2006


STATUS

approved



