Little explorations with HP calculators (no Prime)

[Claudio L.]

(10-31-2017 02:22 PM)pier4r Wrote:  Otherwise I should find the time to explain to myself, somehow, why the +2 distance? Why the +3 distances only after some denominators? Are there more distances? Etc...

Start with the equality. A=number, not necessarily integer. L=Floor(A) and H=ceil(A).

A/n=pi
A=pi*n

Therefore your L and H are:
L=floor(n*pi)
H=ceil(n*pi)

This means in general, L and H both skip in pi steps as you increase n, but after you round with floor/ceil you only move 3 units, leaving a remainder of (pi-3) that accumulates for the next iteration. Once that reminder accumulates to 1 you'll skip 4 instead of 3.
Turns out 1/(pi-3) = 7.06xxxx that's why you accumulate one unit every 7 iterations.