A335505 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 5*x(j-1) by deleting all occurrences of the digit k in base n.

0, 1, 1, 6, 1, 1, 4, 1, 1, 2, 1, 1, 0, 0, 0, 6, 60, 6, 30, 2, 1, 48, 2, 3, 12, 0, 1, 6, 156, 14, 22, 2, 18, 1, 34, 78, 12, 36, 3, 48, 0, 1, 138, 198, 10, 684, 1, 1, 2, 20, 1, 2, 0, 22, 1872, 495, 2, 50, 315, 0, 1, 405, 245, 2780, 0, 1440

Offset

1,4

Comments

T(1,0) = 0 is defined in order to make the triangle of numbers regular.

T(n,k) = 1 whenever k is a power of 5 and k > 1.

Links

Pontus von Brömssen, Rows n = 1..32, flattened

Example

Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9
  -----------------------------------------------------
   1:    0
   2:    1    1
   3:    6    1    1
   4:    4    1    1    2
   5:    1    1    0    0    0
   6:    6   60    6   30    2    1
   7:   48    2    3   12    0    1    6
   8:  156   14   22    2   18    1   34   78
   9:   12   36    3   48    0    1  138  198   10
  10:  684    1    1    2   20    1    2    0   22 1872
T(10,7) = 0 because A335506 never enters a cycle.

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 A335505(n, k):
  return cycle_length(n, k, 5) if n>1 else 0

Crossrefs

Cf. A335502, A335503, A335504, A335506.

Cf. A243846, A306569, A306773.

Sequence in context: A010134 A019713 A155824 * A182429 A325385 A195385

Adjacent sequences: A335502 A335503 A335504 * A335506 A335507 A335508

Keyword

nonn,base,tabl

Author

Pontus von Brömssen, Jun 13 2020

Status

approved