[seqfan] Weak conjecture regarding Viswanath's constant
Trizen
trizenx at gmail.com
Sat Feb 24 22:09:25 CET 2018
Dear SeqFans,
I would like to share with you a weak conjecture regarding Viswanath's
constant ( https://oeis.org/A078416 ). If the conjecture is true, then we
will be able compute the constant relatively easy at arbitrary precision.
First, we consider the following two sequences:
https://oeis.org/A133002 (numerators)
https://oeis.org/A133003 (denominators)
Then we define a(n) as:
a(n) = A133002(n) / A133003(n)
Which is equivalent with:
a(n) = f(n) * n!
where:
f(0) = 1
f(n) = -Sum_{k=0..n-1} f(k) / ((n - k + 1)!)^2
Furthermore, we define:
t = Sum_{k=0..Infinity} 1/a(k)
The decimal expansion of t is:
t = -18497.440650652720515613516713415750667794722534171...
If we enter the absolute value of "t" into Wolfram|Alpha, it gives us the
following formula:
-(4 * (2689 * v - 4875))/(3 * (v - 1)) =
18497.440650652720515613516713415750667794722534171
where "v" is Viswanath's constant.
As we can see here:
http://www.wolframalpha.com/input/?i=18497.440650652720515613516713415750667794722534171
Notice that Wolfram|Alpha highlights all the digits as being correct
(probably a bug? or maybe it just doesn't know Viswanath's constant at high
enough precision?).
However, if Wolfram|Alpha is actually correct, then we can extract
Viswanath's constant from "t", as:
v = (19500 - 3*t)/(10756 - 3*t)
which gives us:
v = 1.1319882487943009051050564211707590973929159046...
Interestingly enough, all the digits of "v" match against A078416, except
for the last two.
Is it possible that "t" is indeed Viswanath's constant? Any thoughts?
Thanks for reading,
Daniel È˜uteu
More information about the SeqFan
mailing list