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.

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, 71, 74, 79, 83, 90, 95, 101, 106, 111, 114, 118, 126, 132, 138, 146, 152, 156, 161, 167, 172, 180, 189, 194, 204, 208, 216, 221, 228, 234, 242, 249, 258, 264, 274, 282

Offset

1,4

Comments

Note that more than one stone can be moved during a single move.

A121924 is the analogous sequence if only one stone can be transferred between piles of equal size.

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).

Both A121924 and A011371 are lower bounds for this sequence.

Links

Table of n, a(n) for n=1..60.

Example

For n = 7, an 8-move sequence is:
(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).

Prog

(Python)
def A292729(n):
    s_in = set([(1, )*n])
    count=-1
    while len(s_in) > 0:
        s_out = set()
        for s in s_in:
            last = -1 ; idx = 0
            while (idx+1) < len(s):
                h = s[idx]
                if h!=last and s[idx+1]==h:
                    for q in range(1, h+1):
                        lst = list(s[:idx]) + list(s[idx+2:])
                        lst += [2*h] if h==q else [ h-q, h+q]
                        t = tuple(sorted(lst))
                        if not t in s_out:
                            s_out.add(t)
                last = s[idx] ; idx += 1
        count += 1
        s_in = s_out
    return count
# Bert Dobbelaere, Jul 14 2019

Crossrefs

Cf. A011371, A121924, A292726, A292728.

Sequence in context: A127735 A330249 A075550 * A328989 A339190 A137529

Adjacent sequences: A292726 A292727 A292728 * A292730 A292731 A292732

Keyword

nonn

Author

Peter Kagey, Sep 22 2017

Extensions

a(35)-a(60) from Bert Dobbelaere, Jul 14 2019

Status

approved