# [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?