cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Jun 13 2020

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A335502, A335503, A335504, A335506.
Cf. A243846, A306569, A306773.

Programs

  • 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

A335503 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 3*x(j-1) by deleting all occurrences of the digit k in base n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 12, 1, 2, 1, 28, 1, 0, 1, 2, 64, 1, 2, 1, 2, 4, 60, 4, 2, 1, 4, 0, 2, 54, 1, 2, 1, 62, 16, 2, 48, 2, 1, 0, 1, 0, 0, 0, 0, 0, 80, 40, 4, 1, 20, 2000, 60, 72, 4, 1, 40, 20, 5, 1, 85, 240, 5, 5, 20, 1, 320, 1260, 128, 2, 1, 272, 4, 2, 48, 68, 1, 20, 1440
Offset: 1

Views

Author

Pontus von Brömssen, Jun 13 2020

Keywords

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 3 and k>1.

Examples

			Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9   10   11
  ---------------------------------------------------------------
   1:    0
   2:    1    1
   3:    1    1    0
   4:   12    1    2    1
   5:   28    1    0    1    2
   6:   64    1    2    1    2    4
   7:   60    4    2    1    4    0    2
   8:   54    1    2    1   62   16    2   48
   9:    2    1    0    1    0    0    0    0    0
  10:   80   40    4    1   20 2000   60   72    4    1
  11:   40   20    5    1   85  240    5    5   20    1  320
  12: 1260  128    2    1  272    4    2   48   68    1   20 1440
T(10,0) = 80, because A243845 eventually enters a cycle of length 80.

Links

Crossrefs

Cf. A243845.
Cf. A335502, A335504, A335505, A335506.
Cf. A243846, A306569, A306773.

Programs

  • 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 A335503(n,k):
  return cycle_length(n,k,3) if n>1 else 0

A335504 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 4*x(j-1) by deleting all occurrences of the digit k in base n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 2, 24, 2, 2, 1, 16, 18, 1, 6, 1, 42, 33, 1, 1, 15, 1, 24, 3, 3, 1, 1, 0, 1, 0, 0, 0, 3, 1, 195, 27, 1, 465, 147, 2, 6, 1002, 18, 4, 42, 1, 66, 2, 10, 10, 738, 1660, 25, 5, 180, 1, 2, 15, 35, 150, 4, 1490
Offset: 1

Views

Author

Pontus von Brömssen, Jun 13 2020

Keywords

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 4 and k>1.

Examples

			Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9
  -----------------------------------------------------
   1:    0
   2:    1    1
   3:    2    1    1
   4:    1    1    0    0
   5:    2   24    2    2    1
   6:   16   18    1    6    1   42
   7:   33    1    1   15    1   24    3
   8:    3    1    1    0    1    0    0    0
   9:    3    1  195   27    1  465  147    2    6
  10: 1002   18    4   42    1   66    2   10   10  738

Links

Crossrefs

Cf. A335502, A335503, A335505, A335506.
Cf. A243846, A306569, A306773.

Programs

  • 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 A335504(n,k):
  return cycle_length(n,k,4) if n>1 else 0
Showing 1-3 of 3 results.

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