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%I #76 Jan 31 2024 07:46:05
%S 1,1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,
%T 7174453,21523360,64570081,193710244,581130733,1743392200,5230176601,
%U 15690529804,47071589413,141214768240,423644304721,1270932914164
%N The 3 X 3 X ... X 3 dots problem (3, n times): minimal number of straight lines (connected at their endpoints) required to pass through 3^n dots arranged in a 3 X 3 X ... X 3 grid.
%C Except for the first term a duplicate of A003462.
%C This is an n-dimensional generalization of the well-known "Nine Dots Problem".
%C Except for n < 2, the a(n) represent "outside the box" solutions, but (for any n) the minimal covering trail C(n) is still inside a box of hyper)-volume 3^n units^n. - _Marco Ripà_, Jul 19 2020
%H M. Ripà, <a href="https://www.researchgate.net/publication/343050221_Solving_the_106_years_old_3k_Points_Problem_with_the_Clockwise-algorithm">Solving the 106 years old 3^k Points Problem with the Clockwise-algorithm</a>, ResearchGate, 2020 (DOI: 10.13140/RG.2.2.34972.92802).
%H M. Ripà, <a href="https://www.researchgate.net/publication/342331014_Solving_the_n_1_n_2_n_3_Points_Problem_for_n_3_6">Solving the n_1 <= n_2 <= n_3 Points Problem for n_3 < 6</a>, ResearchGate, 2020 (DOI: 10.13140/RG.2.2.12199.57769/1).
%H M. Ripà, <a href="http://nntdm.net/volume-20-2014/number-1/59-71/">The rectangular spiral or the n1 X n2 X ... X nk Points Problem</a>, Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Thinking_outside_the_box#Nine_dots_puzzle">Nine dots puzzle</a>
%F a(n) = (3^n - 1)/2 = A003462(n), for n >= 1. - _Marco Ripà_, Jul 19 2020
%e For n=5, a(5) = 121. You cannot touch (the centers of) the 3^5 = 243 points using fewer than 121 straight lines, following the "Nine Dots Puzzle" basic rules.
%t Join[{1}, (3^Range[30]-1)/2] (* _Paolo Xausa_, Jan 31 2024 *)
%Y Cf. A003462, A058992, A225227.
%K nonn
%O 0,3
%A _Marco Ripà_, Aug 24 2015
%E a(4) added by _Marco Ripà_, Aug 06 2018
%E a(3)-a(4) corrected and more terms added by _Marco Ripà_, Jul 19 2020