Yahoo Tuning Groups Ultimate Backup tuning A few truly UNSTABLE Fibonacci-like series

(I'm not really familiar with proper mathematical notation, but the
formulas contained herein should make sense to those who are interested.)

As you might already know, i've done quite a bit of work with various
"Fibonacci-like" series for tuning. The Fibonacci series of course,
leads to the interval of the Golden Mean by the formula

x(n)=x(n-1)+x(n-2)

Other "Fibonacci-like" series i've used are, besides the sometimes
mentioned "tribonacci" series...

x(n)=x(n-1)+x(n-2)+x(n-3)

which can be extended into including four, five factors etc. also
those who have a different pattern of adding the values from
precendents-the second and third last for example:

x(n)=x(n-2)+x(n-3)

or the first and third last.

x(n)=x(n-1)+x(n-3)

Of course there are further varieties of these when picking from the
FOUR latest values. However, all of these additive series clearly
settle on one stable, irrational value-similar to the "original"
Fibonacci series arriving at the golden mean.

I had already considered using subtractive (instead of additive) terms
as part of iterated series, but only recently i tried calculating the
results of them. The results are beyond all expectations.

At first i wouldn't think that noone had found these number series
before, but then it semt very unlikely that i wouldn't have heard of
it, as much as i've read both about abstract and "recreational"
mathematics, and of chaotic algorithms-the relevance of the later you
will see here.

These are truly unpredictable, chaotic number series, generated by a
simple iterated algorithm. But unlike any other such that i've heard
of previously, these deal with INTEGERS.

For example, adding the first preceding number with the negative of
the third last

x(n)=x(n-1)-x(n-3)
^
minus

gives the utterly incoherent series:

1 1 1 0 -1 -2 -2 -1 1 3 4 3 0 -4 -7 -7 -3 4 11 14 10 -1
-15 -25
-24 -9 16 40 49 33 -7 -56 -89 -82 -26 63 145 171 108 -37 -208
-316 -279
-71 245 524 595 350 -174 -769 -1119 -945 -176 943 1888 2064
1121 -767
-2831 -3952 -3185 -354 3598 6783 7137 3539 -3244 -10381 -13920
-10676 -295

And if x(n-1) and x(n-3) are swapped, the result is the same.
Furthermore, having two of the terms negative with one of them positive:

x(n)=x(n-1)-x(n-2)-x(n-3)

It does not matter which one is in what place, the result is the
same-except some of the negative/positive signs are switched:

1 1 0 -2 -3 -1 4 8 5 -7 -20 -18 9 47 56 0 -103 -159 -56 206
421
271 -356 -1048 -963 441 2452 2974 81 -5345 -8400 -3136 10609 22145
14672 -18082 -54899 -51489 21492 127880 157877 8505 -277252 -443634
-174887 545999 1164520 793408 -917111 -2875039 -2751336 1040814
6667189

One last one(still involving only three preceding numbers)is approached by

x(n)=x(n-3)-x(n-2)

x(n)=-x(n-2)-x(n-3)
^
both terms are negative

result being

1 0 -1 1 1 -2 0 3 -2 -3 5 1 -8 4 9 -12 -5 21 -7 -26 28 19
-54 9
73 -63 -64 136 1 -200 135 201 -335 -66 536 -269 -602 805 333
-1407 472
1740 -1879 -1268 3619 -611 -4887 4230 4276 -9117 -46 13393 -9071
-13439
22464 4368 -35903 18096 40271 -53999 -22175 94270 -31824 -116445
126094

I want to know wether anyone has discovered these series before, and
if there are any conclusions to be drawn about the numbers other than
that
"they are chaotic"?

/Mats Öljare

A few truly UNSTABLE Fibonacci-like series

🔗Mats Öljare <oljare@hotmail.com>

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>