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A319259 Minimal number of straight lines of a covering tree needed to cover n X n X n points arranged in a 3-D grid. 0
1, 4, 13, 20 (list; graph; refs; listen; history; text; internal format)



For any n > 2, is it possible to construct an "inside the box" covering tree for any n X n X n set of points consisting of n^2 + n lines if n is even and n^2 + n + 1 lines if n is odd.

In the special case of any 2 X 2 X ... X 2 points problem (k-times n), every optimal covering tree has (exactly) 2^(k-1) rectilinear segments, thus 2^(3-1) = 4 lines for the 2 x 2 x 2 case.

Let n >= 5, we know that it is impossible to cover more than n^3 points with n^2 segments (trivial), and we need at least n segments to obtain a connected graph (otherwise, we cannot join more than n + h*(n - 1) points with h + 1 connected lines). Thus, assuming n > 2, the general lower bound is confirmed to be n^2 + n, therefore a(n even) = 4, 20, 42, 72, ...


Table of n, a(n) for n=1..4.

A. Dumitrescu, C. D. Tóth, Covering Grids by Trees, 26th Canadian Conference on Computational Geometry, 2014.

M. Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem, Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71.


a(n) = n^2 + n for even n.

a(n) = n^2 + n + 1 for odd n (conjectured).


For n = 3 the a(3) = 13 represents the minimum line segments to cover a 3 X 3 X 3 points (symmetrical) grid.


Cf. A058992, A225227, A261547, A318165.

Sequence in context: A141491 A292363 A119048 * A241264 A066993 A297884

Adjacent sequences:  A319256 A319257 A319258 * A319260 A319261 A319262




Marco Ripà, Sep 15 2018



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Last modified September 21 17:46 EDT 2019. Contains 327273 sequences. (Running on oeis4.)