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A261547 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. 3
1, 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453, 21523360, 64570081, 193710244, 581130733, 1743392200, 5230176601, 15690529804, 47071589413, 141214768240, 423644304721, 1270932914164 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..26.

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.

CROSSREFS

Cf. A003462, A058992, A225227.

Sequence in context: A025567 A003462 A076040 * A091141 A098183 A171556

Adjacent sequences:  A261544 A261545 A261546 * A261548 A261549 A261550

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

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Last modified September 20 14:44 EDT 2021. Contains 347586 sequences. (Running on oeis4.)