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.

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

Offset: 1

Author

Hugo Pfoertner, Nov 23 2003

Keywords

• nonn
• tabl
• hard
• more

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.

Examples

			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).
		

References

• For references and links see A087725(n)=T(n,n).

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.

Programs

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

Extensions

T(5,3) copied from A151944 by Hugo Pfoertner, Aug 02 2021