User:Newfraferz87/Sandbox

Wuxia timeline
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A very strange number table
The following is a table that I've "discovered".
 * For any number sequence  a  of position  k , where a is a non-negative integer and k is a natural number:
 * For a=0, ak=0;
 * For a>0, a1=1;
 * (a+1)(k+1) = ak + (a+1)k.

This generates a series of number sequences. The first sequences are the familiar list of natural numbers, Lazy caterer's sequence (bearing a similar resemblance to Triangle numbers) and cake numbers. As any non-first number in a sequence is the sum of the number before it in the same sequence and the number in the same order as the latter in a previous sequence, there will be more numbers being powers of two as the sequence number gets bigger. The "final" sequence ends up with all the powers of two.

What's more interesting is that, for every k where k=2a, a value of 22a-2 (or 4a-1) is returned.

So, I will be really grateful if any kind passerby can tell me the name of this table (if there already is one), or (if possible) the general formula of any particular sequence from the table, such that I can do further research on it.

To dos:

 * Demi-Gods and Semi-Devils: character development with some information from here