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A140518
Number of simple paths from corner to corner of an n X n grid with king-moves allowed.
10
1, 5, 235, 96371, 447544629, 22132498074021, 10621309947362277575, 50819542770311581606906543, 2460791237088492025876789478191411, 1207644919895971862319430895789323709778193, 5996262208084349429209429097224046573095272337986011
OFFSET
1,2
COMMENTS
This graph is the "strong product" of P_n with P_n, where P_n is a path of length n. Sequence A007764 is what we get when we restrict ourselves to rook moves of unit length.
Computed using ZDDs (ZDD = "reduced, order, zero-suppressed binary decision diagram").
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 117, Addison-Wesley, 2009.
LINKS
Jens Weise, Evolutionary Many-Objective Optimisation for Pathfinding Problems, Ph. D. Dissertation, Otto von Guercke Univ., Magdeburg (Germany, 2022), see p. 53.
EXAMPLE
For example, when n=8 this is the number of ways to move a king from a1 to h8 without occupying any cell twice.
CROSSREFS
Main diagonal of A329118.
Cf. A220638 (Hosoya index).
Sequence in context: A265957 A137085 A153025 * A269551 A200962 A270052
KEYWORD
nonn,walk
AUTHOR
Don Knuth, Jul 26 2008
EXTENSIONS
a(9)-a(11) from Andrew Howroyd, Apr 07 2016
STATUS
approved