A151944 Square array read by antidiagonals: T(m,n) = maximal number of moves required for the m X n generalization of the sliding block 15-puzzle (or fifteen-puzzle).

0, 1, 1, 2, 6, 2, 3, 21, 21, 3, 4, 36, 31, 36, 4, 5, 55, 53, 53, 55, 5, 6, 80, 84, 80, 84, 80, 6, 7, 108

Offset: 1

Author

Anton Kulchitsky (kulchits(AT)arsc.edu), Aug 14 2009, Aug 16 2009

Keywords

• nonn
• tabl
• more

Comments

See A087725 for more about this problem and its history.

Examples

			Array begins:
.n\m...1...2...3...4...5...6...7...8...9
.----------------------------------------
.1.|...0...1...2...3...4...5...6...7...8
.2.|...1...6..21..36..55..80.108.140
.3.|...2..21..31..53..84
.4.|...3..36..53..80
.5.|...4..55..84
.6.|...5..80
.7.|...6.108
.8.|...7.140
.9.|...8
		

Links

• Richard Korf, Linear-time Disk-Based Implicit Graph Search, Journal of the ACM 55 (2008), No. 6.
• Anton Kulchitsky, Comments on the Fifteen Puzzle [The entries for the 3x4 puzzle are wrong. The second "8" should be "11" in each of the 18 cases.- _Brian Almond_, Oct 10 2021]

Crossrefs

Main diagonal: A087725. Row 2: A151943.

Cf. A090033 same as this sequence, but written as triangle.

Programs

Python

# alst(), moves(), swap() in A089473
def T(m, n):  # chr(45) is '-'
    start, shape = "".join(chr(45+i) for i in range(m*n)), (m, n)
    return len(alst(start, shape))-1
def auptodiag(maxd):
    for d in range(1, maxd+1):
        for m in range(1, d+1):
            n = d-m+1
            print(T(m, d-m+1), end=", ")
auptodiag(5) # Michael S. Branicky, Aug 02 2021

Extensions

Extensions from Korf's 2008 publication, with corrections, Tomas Rokicki, Aug 17 2011