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Table 11.4 Initial sequence cycle
Position
0
1
2
3
4
5
6
7
8
9
10
11
Number
24
21
15
21
18
24
18
15
21
27
|12
18
Table 11.5 Sequence cycle
Position
10
11
0
1
2
3
4
5
6
7
8
9
Number
12
18
24
21
15
21
18
24
18
15
21
27
11.4.2
PERIODIC
This is an oscillation in the differential coefficient shift of a sequence where POS min
is the position of the smallest number in the sequence and LCD i is the Least Common
Divisor. The function 'mod' is taken in the programming sense of being the remainder
after integer division.
(( i
Sub-harmonic function
=
POS min ) mod LCD i )
Example:
Here we highlight the start of the cycle with the symbol '|'. The complete cycle
and the initial positions is shown in Table 11.4
( 24, 21, 15, 21, 18, 24, 18, 15, 21, 27,
|
12, 18, 24 , 21 , 15 , 21 , 18 )
The start is found by looking for the minimum value (i.e. 12). The numbers after
the pair (12, 18), in bold italic, match the numbers in italic at the beginning of
the sequence. So, this sub-sequence starting at 12 represents the initial part of the
complete cycle. So doing 'a right shift circular seven times' on the numbers (an old
style computer operation applied to bits in a computer word) we have:
|
12 , 18 , 24, 21, 15, 21, 18 )( 24, 21, 15, 21, 18, 24 , 18 , 15 , 21 , 27 ,
So the sub sequence in italic 24, 21, 15, 21, 18 is the same as the italic 24, 21, 15,
21, 18 at the start of the sequence. One of these sub-sequences can be deleted to give
the minimum cyclic component of the total sequence as in Table 11.5
Taking the original given sequence starting with 24, 18, the cyclic component
starts at position 10 (numbering from 0). The length of the cycle is 12. This has the
least common divisors (LCD) of (3, 4). Using the Sub-harmonic function above we
can express another formula for this periodic sequence as a function of the index i
as (Table 11.6 ):
K (( i
Pf ( i )
=
S Min +
POS min ) mod LCD i )
S Min =
12 - > minimum value in sequence
POS min =
10 - > position/index of minimum value
 
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