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A090033
Triangle T(j,k) read by rows, where T(j,k) is the number of single tile moves in the longest optimal solution of the j X k generalization of the sliding block 15-puzzle, starting with the empty square in a corner.
10
0, 1, 6, 2, 21, 31, 3, 36, 53, 80, 4, 55, 84
OFFSET
1,3
COMMENTS
T(k,j) = T(j,k).
T(2,2), T(2,3), T(4,2), T(4,3) from Karlemo and Östergård, T(3,3) from Reinefeld, T(4,4) from Bruengger et al.
REFERENCES
For references and links see A087725(n)=T(n,n).
EXAMPLE
The triangle begins
0
1 6
2 21 31
3 36 53 80
4 55 84 ...
.
a(6)=T(3,3)=31 because the A090163(3,3)=2 longest optimal solution paths of the 3 X 3 (9-) sliding block puzzle have length 31 (see A089473).
PROG
(Python) # alst(), moves(), swap() in A089473
def T(j, k): # chr(45) is '-'
start, shape = "".join(chr(45+i) for i in range(j*k)), (j, k)
return len(alst(start, shape))-1
for j in range(1, 5):
for k in range(1, j+1):
print(T(j, k), end=", ") # Michael S. Branicky, Aug 02 2021
CROSSREFS
Cf. A087725, A089473, A089484, A090034, A090035, A090036, A090166, A090163 corresponding number of different configurations with largest distance.
Cf. A151944 same as this sequence, but written as full array.
Sequence in context: A213503 A169632 A201445 * A036173 A142707 A305874
KEYWORD
nonn,tabl,hard,more
AUTHOR
Hugo Pfoertner, Nov 23 2003
EXTENSIONS
T(5,3) copied from A151944 by Hugo Pfoertner, Aug 02 2021
STATUS
approved