My favorite integer sequence

[jaydfox]

I have the feeling Jay's post/idea is not complete yet or still under development , probably there will be an edit or a followup.

A little bit strange notation I think , but that might be due to the unfinished brainstorming too.

Yes, still brainstorming, and I'm not attached to any particular notation yet. But I needed something so I could start to formalize...

My first critical remark is that the analogue for exp does not seem convincing right now ; I do not see (1+1/n)^n.
But I might need to read again and Jay will probably do an edit or followup. Its just a first impression anyway.

(...)

I was referring to Jay's post 30 mainly. He posted nr 31 while I made this reply.

Take exp_1:

Code:

k    E_1(k)    E_{1/2}(k)   E_{1/3}(k)
0    1         1            1
1    2         3/2          4/3
2    4         9/4          16/9
3    8         27/4         64/27

exp_1(1) = 2 = (1+1)^1
exp_{1/2}(1) = 2.25 = (1+1/2)^2
exp_{1/3}(1) = 2.370370... = (1+1/3)^3
exp_{1/n}(1) = (1+1/n)^n

I could have defined exp_h that way to begin with, but I defined it as a sequence with a recurrence relation similar to A(k), to help draw the analogy between a_1 = 1.08306 and e_1= e/2 = 1.35914.

Considering your lastest posts Jay , are you still convinced that J(x)/1.08... is a very good approximation to A(x) ?

The reason I wanted to set 1.08306 as a_1 was to make clear what it was. By itself, J(x) is not a perfect approximation of A(k). To Tommy's question, do I consider J(x) a "very good approximation" to A(x)? A "good approximation", yes, but not a "very good approximation".

But by re-imagining A(k) as A_1(k), it makes clear why it's not such a good approximation. In reality, it's not so much an issue of J(x) being a poor approximation of A(k). It's that A_1(k) uses such a large step size, that it's a poor approximation of J(x). In this sense, J(x) is the right function to approximate A_1(k), but not the right function to approximate A(k). The distinction is not mathematical, but metaphysical.