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.

0, 2, 2, 6, 60, 6, 12, 1058, 1058, 12, 20, 25080, 140240, 25080, 20, 30, 822594, 58673472, 58673472, 822594, 30, 42, 36195620, 28938943114, 490225231968, 28938943114, 36195620, 42, 56, 2069486450

Offset

1,2

Comments

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

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

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

Links

Table of n, a(n) for n=1..30.

StackExchange, Combination of smartphones' pattern password, 2014

Example

G(2,2) is the complete graph on 4 vertices, hence T(2, 2) = 4*3 + 4*3*2 + 4*3*2*1 = 60.
T(n, k) for n + k <= 8 is as follows:
.0........2...........6...........12..........20.......30..42
.2.......60........1058........25080......822594.36195620
.6.....1058......140240.....58673472.28938943114
12....25080....58673472.490225231968
20...822594.28938943114
30.36195620
42

Crossrefs

Cf. A247944.

Sequence in context: A284707 A174589 A326942 * A329571 A270358 A156529

Adjacent sequences: A247940 A247941 A247942 * A247944 A247945 A247946

Keyword

nonn,tabl

Author

Rob Arthan, Sep 27 2014

Status

approved