**Appendix 5 - An Introduction to
Midpoint Sequences**

At the
centre of every *odd* number (N, say) lies another (M,
say) - where M = (N+1)/2. Clearly, M may be *odd* or *even*.
If odd, then a sequence of two or more such *centres *may
be obtained by repeating this procedure on successive outcomes
until an even number appears to terminate the sequence.

As an illustration, consider the case in which N = 833. This leads to the following sequence of 6 midpoints:

833 ----> 417 ----> 209 ----> 105 ----> 53 ----> 27 ----> 14

From an
analytical point of view this sequence is better viewed *in
reverse*, thus:

14 ----> 27 ----> 53 ----> 105 ----> 209 ----> 417 ----> 833 ----> 1665 ----> 3329 ----> ... without limit

From
this we learn that in choosing 833 as our starting value we were
actually breaking into an *infinite* sequence in which
each term after the first is double the previous term, less 1.
Clearly, what we may conveniently term *the root *of such
a sequence is some even number (E, say) - the sequence itself may
then be referred to as S(E), so that in the foregoing example we
view the opening terms of S(14).

We may represent the general sequence as

S(E) = M(0), M(1), M(2), ..., M(R) ..., M(R+1), ... to infinity, where the root of the sequence, M(0) = E.

Should
any of the terms happen to be a triangular number then the next
following term may be represented geometically as an * hour-glass*.
As an example, consider M(3) of the sequence S(14) above. Its
value is 105 - 14th triangular number. This is depicted in the
following diagram along with M(4), or 209, in hour-glass form.

Vernon Jenkins MSc

2005-4-23