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Number of paths of length n+2 originating at a corner of a 4 X 4 Boggle board.
3

%I #9 Mar 16 2021 04:10:07

%S 3,15,75,322,1184,3814,10918,27772,61734,116966,183256,228016,211502,

%T 128994,37948

%N Number of paths of length n+2 originating at a corner of a 4 X 4 Boggle board.

%C A legal walk on a Boggle board is from one vertex to a vertex adjacent orthogonally or diagonally.

%H Eugene McDonnell, <a href="/A063000/a063000.txt">Remarks on Game of "Boggle"</a>

%F Formula unknown, values computed empirically.

%o (Python)

%o def nn(c): # neighbors of c

%o i, j = divmod(c, 4)

%o N = set((i+io, j+jo) for io in [-1, 0, 1] for jo in [-1, 0, 1]) - {(i, j)}

%o return [4*i+j for i, j in N if 0 <= i < 4 and 0 <= j < 4]

%o def afind():

%o n, paths = 0, {(0,)}

%o while n+2 <= 16:

%o paths1 = set(p + (e,) for p in paths for e in nn(p[-1]) if e not in p)

%o print(len(paths1), end=", ")

%o n, paths = n+1, paths1

%o afind() # _Michael S. Branicky_, Mar 15 2021

%Y Cf. A063001, A063002.

%K fini,full,nonn

%O 0,1

%A Eugene McDonnell (EEMcD(AT)AOL.com), Jul 01 2001