

A151944


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


2



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
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OFFSET

1,4


COMMENTS

See A087725 for more about this problem and its history.


LINKS

Table of n, a(n) for n=1..30.
Richard Korf, Lineartime DiskBased 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]


EXAMPLE

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


PROG

# (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 = dm+1
print(T(m, dm+1), end=", ")
auptodiag(5) # Michael S. Branicky, Aug 02 2021


CROSSREFS

Main diagonal: A087725. Row 2: A151943.
Cf. A090033 same as this sequence, but written as triangle.
Sequence in context: A334188 A265993 A115009 * A073094 A194953 A057606
Adjacent sequences: A151941 A151942 A151943 * A151945 A151946 A151947


KEYWORD

nonn,tabl,more


AUTHOR

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


EXTENSIONS

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


STATUS

approved



