OFFSET
0,3
COMMENTS
Except for the first term a duplicate of A003462.
This is an n-dimensional generalization of the well-known "Nine Dots Problem".
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
LINKS
M. Ripà, Solving the 106 years old 3^k Points Problem with the Clockwise-algorithm, ResearchGate, 2020 (DOI: 10.13140/RG.2.2.34972.92802).
M. Ripà, Solving the n_1 <= n_2 <= n_3 Points Problem for n_3 < 6, ResearchGate, 2020 (DOI: 10.13140/RG.2.2.12199.57769/1).
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.
Wikipedia, Nine dots puzzle
FORMULA
a(n) = (3^n - 1)/2 = A003462(n), for n >= 1. - Marco Ripà, Jul 19 2020
EXAMPLE
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.
MATHEMATICA
Join[{1}, (3^Range[30]-1)/2] (* Paolo Xausa, Jan 31 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Marco Ripà, Aug 24 2015
EXTENSIONS
a(4) added by Marco Ripà, Aug 06 2018
a(3)-a(4) corrected and more terms added by Marco Ripà, Jul 19 2020
STATUS
approved