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 A247944 2-dimensional array T(n, k) listed by antidiagonals for n >= 2, k >= 1 giving the number of acyclic paths of length k in the graph G(n) whose vertices are the integer lattice points (p, q) with 0 <= p, q < n and with an edge between v and w iff the line segment [v, w] contains no other integer lattice points. 1
 12, 24, 56, 24, 304, 172, 0, 1400, 1696, 400, 0, 5328, 15580, 6072, 836, 0, 16032, 132264, 88320, 18608, 1496, 0, 35328, 1029232, 1225840, 403156, 44520, 2564, 0, 49536, 7286016, 16202952, 8471480, 1296952, 100264, 4080, 0, 32256, 46456296, 203422072, 172543276, 36960168, 3864332, 201992, 6212 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS G(3) is used for Android screen lock security patterns (see StackExchange link). There is an edge between v = (p, q) and w = (r, s) iff p - r and q - s are coprime. T(n, k) is nonzero for 1 <= k < n^2 and is zero for k >= n^2, because G(n) always has an acyclic path that contains all n^2 vertices and hence has length n^2 - 1, while a path in G(n) of length n^2 or more cannot be acyclic. The row sums of this sequence form the nonzero entries on the diagonal of A247943. LINKS StackExchange, Combination of smartphones' pattern password, 2014. EXAMPLE In G(3), the 4 vertices at the corners have valency 5, the vertex in the middle has valency 8 and the other 4 vertices have valency 7, therefore T(3, 2) = 4*5*4 + 8*7 + 4*7*6 = 304. T(n, k) for n + k <= 11 is as follows: ..12.....24......24........0.........0.........0........0.....0.0 ..56....304....1400.....5328.....16032.....35328....49536.32256 .172...1696...15580...132264...1029232...7286016.46456296 .400...6072...88320..1225840..16202952.203422072 .836..18608..403156..8471480.172543276 1496..44520.1296952.36960168 2564.100264.3864332 4080.201992 6212 T(4, k) is nonzero iff k <= 15 and the 15 nonzero values are: 172, 1696, 15580, 132264, 1029232, 7286016, 46456296, 263427744, 1307755352, 5567398192, 19756296608, 56073026336, 119255537392, 168794504832, 119152364256. The sum of these 15 values is A247943(4, 4). - Rob Arthan, Oct 19 2014 CROSSREFS Cf. A247943. Sequence in context: A260261 A080495 A090776 * A123980 A097704 A289132 Adjacent sequences:  A247941 A247942 A247943 * A247945 A247946 A247947 KEYWORD nonn,tabl AUTHOR Rob Arthan, Sep 27 2014 STATUS approved

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Last modified June 20 15:46 EDT 2021. Contains 345165 sequences. (Running on oeis4.)