OFFSET
1,4
COMMENTS
T(1,0) = 0 is defined in order to make the triangle of numbers regular.
One way of getting T(n,k) = 0 is to have x(j) = x(i)*n^e for some j > i and e > 0. For k < n <= 48, this is the only way to get T(n,k) = 0 (but see A335506 for another situation where the x-sequence is not periodic).
T(n,k) = 1 whenever k is a power of 2 and k > 1.
It seems that k = 0 and k = n-1 often lead to particularly long cycles.
LINKS
Pontus von Brömssen, Rows n = 1..48, flattened
EXAMPLE
Triangle begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11
---------------------------------------------------
1: 0
2: 1 1
3: 4 1 1
4: 2 1 1 0
5: 4 1 1 3 1
6: 12 2 1 6 1 4
7: 78 1 1 6 1 3 6
8: 3 1 1 0 1 0 0 0
9: 6 1 1 18 1 4 36 4 1
10: 36 4 1 4 1 8 4 72 1 540
11: 100 1 1 16 1 4 17 0 1 8 4
12: 90 2 1 12 1 4 14 6 1 4 4 240
For n = 10 and k = 5, the x-sequence starts 1, 2, 4, 8, 16, 32, 64, 128, 26, 2, and then repeats with a period of 8, so T(10,5) = 8.
T(10,0) = 36, because A242350 eventually enters a cycle of length 36.
For n=11 and k=7, the x-sequence starts (in base 11) 1, 2, 4, 8, 15, 2A, 59, 10. Disregarding trailing zeros, the sequence then repeats with period 7 and x(i+7*j) = x(i)*11^j for positive i and j. The x-sequence itself is therefore not eventually periodic, so T(11,7)=0.
PROG
(Python)
from sympy.ntheory.factor_ import digits
from functools import reduce
def drop(x, n, k):
# Drop all digits k from x in base n.
return reduce(lambda x, j:n*x+j if j!=k else x, digits(x, n)[1:], 0)
def cycle_length(n, k, m):
# Brent's algorithm for finding cycle length.
# Note: The function may hang if the sequence never enters a cycle.
if (m, n, k)==(5, 10, 7):
return 0 # A little cheating; see A335506.
p=1
length=0
tortoise=hare=1
nz=0
while True:
hare=drop(m*hare, n, k)
while hare and hare%n==0:
hare//=n
nz+=1 # Keep track of the number of trailing zeros.
length+=1
if tortoise==hare:
break
if p==length:
tortoise=hare
nz=0
p*=2
length=0
return length if not nz else 0
def A335502(n, k):
return cycle_length(n, k, 2) if n>1 else 0
CROSSREFS
KEYWORD
AUTHOR
Pontus von Brömssen, Jun 13 2020
STATUS
approved