Let's first define 3 parameters (m,i,j) and one variable (n).
As an example, we'll set i=10,j=1, and vary m over it's full cycle modulo i^j (in this case that means 0≤m<10).
Next, we generate a sequence by iterating n in the following expression until it becomes cyclical.
With our example values, we get the following:
Code:
Now, in full generality (for the first 10 diagonals of 2-tuples):
Code:
I have about 38MB of data for analysis from running this, lets see what you guys come up with.
As an example, we'll set i=10,j=1, and vary m over it's full cycle modulo i^j (in this case that means 0≤m<10).
Next, we generate a sequence by iterating n in the following expression until it becomes cyclical.
With our example values, we get the following:
Code:
0: 1; 0 (this yields a lead-in of length 1 and a cycle of length 1)
1: 1 (this leads a cycle of length 1)
2: 1; 2, 4, 6, 8, ... (this yields a lead-in of length 1, and a cycle of length 4)
3: 1, 3, 9, 7, ... (this yields a cycle of length 4)
4: 1; 4, 6, ... (lead-in of length 1, and cycle of length 2)
5: 1; 5, ... (lead-in of length 1, and cycle of length of length 1)
6: 1; 6, ... (lead-in of length 1 and cycle of length 0)
7: 1, 7, 9, 3, ... (this yields a cycle of length 4)
8: 1; 8, 4, 2, 6, ... (this yields a lead-in of length 1, and a cycle of length 4)
9: 1, 9, ... (this yields a cycle of length 2)
Now, in full generality (for the first 10 diagonals of 2-tuples):
Code:
#!/usr/bin/env python
from math import sqrt
def inv_tri(t):
return 0.5*(sqrt(8*t+1)-1)
def tri(n):
return n*(n+1)/2
dct = {}
props = {}
for i in range(tri(11)):
it = inv_tri(i)
first = int(it)
second = i - tri(first)
pair = (first - second, second)
base,exponent = pair
modulus = base**exponent
if modulus: print pair
for j in range(modulus):
id = (j, base, exponent)
num = 1
dct[id] = []
while num not in dct[id]:
dct[id].append(num)
num = (num*j) % modulus
run_up = dct[id].index(num)
total = len(dct[id])
cycle = total - run_up
props[id] = (run_up,cycle,total)
print "\t%d (of %d)" % (j, modulus)
print "\t", dct[id]
print "\t", props[id]
print ""
I have about 38MB of data for analysis from running this, lets see what you guys come up with.
