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A247943 2-dimensional array T(n, k) listed by antidiagonals giving the number of acyclic paths in the graph G(n, k) whose vertices are the integer lattice points (p, q) with 0 <= p < n and 0 <= q < k and with an edge between v and w iff the line segment [v, w] contains no other integer lattice points. 1

%I

%S 0,2,2,6,60,6,12,1058,1058,12,20,25080,140240,25080,20,30,822594,

%T 58673472,58673472,822594,30,42,36195620,28938943114,490225231968,

%U 28938943114,36195620,42,56,2069486450

%N 2-dimensional array T(n, k) listed by antidiagonals giving the number of acyclic paths in the graph G(n, k) whose vertices are the integer lattice points (p, q) with 0 <= p < n and 0 <= q < k and with an edge between v and w iff the line segment [v, w] contains no other integer lattice points.

%C There is an edge between v = (p, q) and w = (r, s) iff p - r and q - s are coprime.

%C G(3, 3) is used for Android screen lock security patterns (see StackExchange link).

%C The nonzero entries on the diagonal of this sequence comprise the row sums of A247944.

%H StackExchange, <a href="http://math.stackexchange.com/questions/37167/combination-of-smartphones-pattern-password">Combination of smartphones' pattern password</a>, 2014

%e G(2,2) is the complete graph on 4 vertices, hence T(2, 2) = 4*3 + 4*3*2 + 4*3*2*1 = 60.

%e T(n, k) for n + k <= 8 is as follows:

%e .0........2...........6...........12..........20.......30..42

%e .2.......60........1058........25080......822594.36195620

%e .6.....1058......140240.....58673472.28938943114

%e 12....25080....58673472.490225231968

%e 20...822594.28938943114

%e 30.36195620

%e 42

%Y Cf. A247944.

%K nonn,tabl

%O 1,2

%A _Rob Arthan_, Sep 27 2014

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)