Fairshare between two and more

Fairshare between two and more is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The Thue-Morse sequence is a sequnce of ones and zeros that if two people take turns in the given order, the first persons turn for every '0' in the sequence, the second for every '1'; then this is shown to give a fairer, more equitable sharing of resources. (Football penalty shoot-outs for example, might not favour the team that goes first as much if the penalty takers take turns according to the Thue-Morse sequence and took 2^n penalties)

The Thue-Morse sequence of ones-and-zeroes can be generated by:

"When counting in binary, the digit sum modulo 2 is the Thue-Morse sequence"

Sharing fairly between two or more

Use this method:

When counting base b, the digit sum modulo b is the Thue-Morse sequence of fairer sharing between b people.

Task

Counting from zero; using a function/method/routine to express an integer count in base b, Sum the digits modulo b to produce the next member of the Thue-Morse fairshare series for b people.

Show the first 25 terms of the fairshare sequence

• For two people
• For three people
• For five people
• For seven people

References

• Non-decimal radices/Convert
• Thue-Morse
• A010060, A053838, A053840, A053842: The On-Line Encyclopedia of Integer Sequences® (OEIS®)

Python

<lang python>from itertools import count, islice

def _basechange_int(num, b):

   """
   Return list of ints representing positive num in base b

   >>> b = 3
   >>> print(b, [_basechange_int(num, b) for num in range(11)])
   3 [[0], [1], [2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [1, 0, 0], [1, 0, 1]]
   >>>
   """
   if num == 0:
       return [0]
   result = []
   while num != 0:
       num, d = divmod(num, b)
       result.append(d)
   return result[::-1]

def fairshare(b=2):

   for i in count():
       yield sum(_basechange_int(i, b)) % b

if __name__ == '__main__':

   for b in (2, 3, 5, 7):
       print(f"{b:>2}: {str(list(islice(fairshare(b), 25)))[1:-1]}")</lang>

Output:

 2: 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0
 3: 0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1
 5: 0, 1, 2, 3, 4, 1, 2, 3, 4, 0, 2, 3, 4, 0, 1, 3, 4, 0, 1, 2, 4, 0, 1, 2, 3
 7: 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 0, 2, 3, 4, 5, 6, 0, 1, 3, 4, 5, 6