[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!


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)) =

where "v" is Viswanath's constant.

As we can see here:

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 Șuteu

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