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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.

%I A090033 #24 Aug 02 2021 06:41:03
%S A090033 0,1,6,2,21,31,3,36,53,80,4,55,84
%N 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.
%C A090033 T(k,j) = T(j,k).
%C A090033 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.
%D A090033 For references and links see A087725(n)=T(n,n).
%e A090033 The triangle begins
%e A090033   0
%e A090033   1   6
%e A090033   2  21  31
%e A090033   3  36  53  80
%e A090033   4  55  84  ...
%e A090033 .
%e A090033 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).
%o A090033 (Python) # alst(), moves(), swap() in A089473
%o A090033 def T(j, k):  # chr(45) is '-'
%o A090033     start, shape = "".join(chr(45+i) for i in range(j*k)), (j, k)
%o A090033     return len(alst(start, shape))-1
%o A090033 for j in range(1, 5):
%o A090033     for k in range(1, j+1):
%o A090033         print(T(j,k), end=", ") # _Michael S. Branicky_, Aug 02 2021
%Y A090033 Cf. A087725, A089473, A089484, A090034, A090035, A090036, A090166, A090163 corresponding number of different configurations with largest distance.
%Y A090033 Cf. A151944 same as this sequence, but written as full array.
%K A090033 nonn,tabl,hard,more
%O A090033 1,3
%A A090033 _Hugo Pfoertner_, Nov 23 2003
%E A090033 T(5,3) copied from A151944 by _Hugo Pfoertner_, Aug 02 2021