[seqfan] sequences defined by period mod n of a map iteration ?
Georgi Guninski
guninski at guninski.com
Tue Oct 12 18:48:30 CEST 2010
define sequence b(i) mod n:
b(1)= INITIAL_B mod n
if condition(b_i) then b(i+1)=f1(b(i)) mod n else b(i+1)=f2(b(i)) mod n
b(i) must be periodic mod n because of the box principle (possibly having a pre-period or reaching a fixed point).
define a(n) to be the period of b(i) mod n (not counting the pre-period).
examples:
[A]
INITIAL_B=2
if b(i)>=floor(n/2) b(i+1)=2*b(i)+1 mod n else b(i+1)=2*b(i)-1 mod n
a(n) is the period of b(i) mod n:
1, 1, 2, 3, 2, 2, 4, 2, 2, 8, 3, 4, 3, 8, 2, 5, 4, 3, 10, 11, 2, 13, 12, 6, 3, 23, 4, 5, 4, 10, 8, 12, 2, 10, 18, 12, 4, 20, 3, 14, 10, 4, 11, 23, 2, 21, 20, 8, 12, 35, 6, 20, 3, 18, 28, 8, 4, 54, 5, 6, 5, 12, 10, 27, 8, 11, 12, 35, 2, 9, 36, 20, 18, 30, 12, 39, 4, 18, 20, 10, 3, 8, 14, 28, 10, 11, 4, 12, 11, 5, 23, 36, 2, 29, 21, 10, 20, 31
no fixed points >3 <12000
[B]
INITIAL_B=2
if b(i)<=floor(n/2) b(i+1)=2*b(i)+1 mod n else b(i+1)=2*b(i)-1 mod n
has fixed points, satisfies a(n+1)=a(2*n)
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