Re: recursive sequence

In article <w6Pqi.4796$RQ5.1231@xxxxxxxxxxxxxxxxxxxxxx>, "Dana" <ddelouis@xxxxxxxxxxxxx> wrote:

[ re: a[k] = a[k-1] * (a[k-1] - 1) ]

!! Just two Cents. Some references use a[0]==3 instead of a[1]==3.
!!
!! http://www.research.att.com/~njas/sequences/A117805
!!
!! Could have sword there would be a closed form, but apparently not.

it seems an interesting sequence that might have some pretty little secrets

here are some easy properties

if m < n, a[m] | a[n]
k
2
we have a[k] = O(3 ) and the 3 can be relaxed

we can investigate prime factors by checking iteration mod p

generally
a[k] = 0 (mod p) -> a[k+1] = 0 (mod p)
a[k] = 1 (mod p) -> a[k+1] = 0 (mod p)
a[k] = 2 (mod p) -> a[k+1] = 2 (mod p)
a[k] = -1 (mod p) -> a[k+1] = 2 (mod p)

for the given sequence we are interested in 3 (mod p)
but more generally we can look at a[0] (mod p) for various a[0]
and find which p eventually become factors

specifically for 5 to 19 we have

5
-
3 -> 1 -> 0

7
-
3 -> 6 -> 2
4 -> 5 -> 6 -> 2
5 -> 6 -> 2

11
--
3 -> 6 -> 8 -> 1 -> 0
4 -> 1 -> 0
5 -> 9 -> 6 -> 8 -> 1 -> 0
6 -> 8 -> 1 -> 0
7 -> 9 -> 6 -> 8 -> 1 -> 0
8 -> 1 -> 0
9 -> 6 -> 8 -> 1 -> 0

13
--
3 -> 6 -> 4 -> 12 -> 2
4 -> 12 -> 2
5 -> 7 -> 3 -> 6 -> 4 -> 12 -> 2
6 -> 4 -> 12 -> 2
7 -> 3 -> 6 -> 4 -> 12 -> 2
8 -> 4 -> 12 -> 2
9 -> 7 -> 3 -> 6 -> 4 -> 12 -> 2
10 -> 12 -> 2
11 -> 6 -> 4 -> 12 -> 2

17
--
3 -> 6 -> 13 -> 3
4 -> 12 -> 13 -> 3
5 -> 3
6 -> 13 -> 3
7 -> 8 -> 3
8 -> 5 -> 3
9 -> 4 -> 12 -> 13 -> 3
10 -> 5 -> 3
11 -> 8 -> 3
12 -> 13 -> 3
13 -> 3
14 -> 12 -> 13 -> 3
15 -> 6 -> 13 -> 3

19
--
3 -> 6 -> 11 -> 15 -> 1 -> 0
4 -> 12 -> 18 -> 2
5 -> 1 -> 0
6 -> 11 -> 15 -> 1 -> 0
7 -> 4 -> 12 -> 18 -> 2
8 -> 18 -> 2
9 -> 15 -> 1 -> 0
10 -> 14 -> 11 -> 15 -> 1 -> 0
11 -> 15 -> 1 -> 0
12 -> 18 -> 2
13 -> 4 -> 12 -> 18 -> 2
14 -> 11 -> 15 -> 1 -> 0
15 -> 1 -> 0
16 -> 12 -> 18 -> 2
17 -> 6 -> 11 -> 15 -> 1 -> 0

so we are able to see a number of different behaviors

all non-[0, 1, 2, -1] go to
all 0s
all 2s
a cycle around 3
some go to 0 and some 2
...

cycles and fixed points in many imaginable execution/iteration digraphs

much more complicated than even pochhammers and related

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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