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A292729 a(n) is the maximum number of steps that can occur during the following procedure: start with n piles each containing one stone; any number of stones can be transferred between piles of equal size.

%I A292729 #25 Nov 05 2024 18:56:52
%S A292729 0,1,1,3,4,4,8,10,11,12,16,20,22,24,27,31,35,38,43,45,47,52,57,62,67,
%T A292729 71,74,79,83,90,95,101,106,111,114,118,126,132,138,146,152,156,161,
%U A292729 167,172,180,189,194,204,208,216,221,228,234,242,249,258,264,274,282
%N A292729 a(n) is the maximum number of steps that can occur during the following procedure: start with n piles each containing one stone; any number of stones can be transferred between piles of equal size.
%C A292729 Note that more than one stone can be moved during a single move.
%C A292729 A121924 is the analogous sequence if only one stone can be transferred between piles of equal size.
%C A292729 A011371 is the analogous sequence if all stones must be transferred between piles of equal size (i.e., the number of stones in each pile must be a power of two).
%C A292729 Both A121924 and A011371 are lower bounds for this sequence.
%e A292729 For n = 7, an 8-move sequence is:
%e A292729 (1 1 1 1 1 1 1) -> (2 1 1 1 1 1) -> (2 2 1 1 1) -> (3 1 1 1 1) -> (3 2 1 1) -> (3 2 2) -> (3 3 1) -> (5, 1, 1) -> (5 2).
%o A292729 (Python)
%o A292729 def A292729(n):
%o A292729     s_in = set([(1, )*n])
%o A292729     count=-1
%o A292729     while len(s_in) > 0:
%o A292729         s_out = set()
%o A292729         for s in s_in:
%o A292729             last = -1 ; idx = 0
%o A292729             while (idx+1) < len(s):
%o A292729                 h = s[idx]
%o A292729                 if h!=last and s[idx+1]==h:
%o A292729                     for q in range(1, h+1):
%o A292729                         lst = list(s[:idx]) + list(s[idx+2:])
%o A292729                         lst += [2*h] if h==q else [ h-q, h+q]
%o A292729                         t = tuple(sorted(lst))
%o A292729                         if not t in s_out:
%o A292729                             s_out.add(t)
%o A292729                 last = s[idx] ; idx += 1
%o A292729         count += 1
%o A292729         s_in = s_out
%o A292729     return count
%o A292729 # _Bert Dobbelaere_, Jul 14 2019
%Y A292729 Cf. A011371, A121924, A292726, A292728.
%K A292729 nonn
%O A292729 1,4
%A A292729 _Peter Kagey_, Sep 22 2017
%E A292729 a(35)-a(60) from _Bert Dobbelaere_, Jul 14 2019