Absolutely cool thing I learnt today. There's an IIT Bombay Grad, University of Michigan resident prof by the name Divakar Vishwanath.

He has a mathematical constant in his name.

Like Planck's constant, Boltzmann constant, Euler's constant etc, there is a mathematical constant in the name of this Divakar Vishwanath called Vishwanath's constant.

I came to know about him from a Numberphile video on YouTube.

His constant is related to Randomised Fibonacci sequences.

So to the uninitiated, Fibonacci sequences are such that any value starting from 1,1 will be such that it is the sum of the previous two values.

So the next value here is 2.

It goes on as 1,1,2, 3, 5, 8, 13, 21, 34, 55 etc. I guess you get the drift. Such a Fibonacci sequence will converge to a very popular number called the Golden ratio which is (1 + √5)/2 or 1.61803399...
This is the ratio of any given value in the sequence to its previous one.

Randomised Fibonacci sequence which Dr. Vishwanath has worked on is slightly different from the Fibonacci sequence.

Here any value R_n will be R_n-1 ± R_n-2. So it will be the addition or subtraction of the previous value with 50% probability - a toss of a coin.

Let's say heads means you add and tails means you subtract. Then, if we start with 1,1, and you get a head the next value will be 2. Now you get a tail, you subtract 1 from 2 to get 1. Then you get another tail you get -1 and etc to get the series 1,1,2, 1, -1 and so on.

The above sequence is just one of the kings of sequences possible with randomised Fibonacci sequences. Depending upon what appears on the coin toss, different kinds of sequences can appear in this scenario. That's why it's called randomised Fibonacci sequence.

So if you get HHHTTTHT starting from 1,1 the sequence will go something like this:

1,1, 2(H), 3(H), 5(H), 2(T), -3 (T), -5(T), -8(H), -3(T) etc.

The question is what is the ratio of any number to previous number in this sequence.

Just like golden ratio is there any ratio for such a sequence was the question Vishwanath answered. Turns out there is such a ratio and the ratio is called Vishwanath's constant and it's equal to 1.319882487943...

This is the number in a randomised Fibonacci sequence which is the ratio of any value in the sequence to the previous value in the same sequence!!

Wow!! Nice info for the day!!

The first photo here is Dr. Vishwanath and the second photo here is his paper on Random Fibonacci sequence ratios.

Let's make him famous guys!!