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A223332 Rolling cube footprints: number of 3 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge. 1

%I #10 Aug 19 2018 10:42:02

%S 64,243,3969,64827,1059723,17324685,283231809,4630406067,75700050363,

%T 1237579942725,20232537609249,330771018512907,5407599817782603,

%U 88405979220238365,1445302430884160289,23628482316968449347

%N Rolling cube footprints: number of 3 X n 0..7 arrays starting with 0 where 0..7 label vertices of a cube and every array movement to a horizontal or antidiagonal neighbor moves along a corresponding cube edge.

%C Row 3 of A223331.

%H R. H. Hardin, <a href="/A223332/b223332.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 18*a(n-1) - 27*a(n-2) for n>4.

%F Conjectures from _Colin Barker_, Aug 19 2018: (Start)

%F G.f.: x*(64 - 909*x + 1323*x^2 - 54*x^3) / (1 - 18*x + 27*x^2).

%F a(n) = (49/54)*((9-3*sqrt(6))^n + (3*(3+sqrt(6)))^n) for n>2.

%F (End)

%e Some solutions for n=3:

%e 0 2 6 0 4 6 0 2 6 0 1 5 0 2 3 0 1 5 0 2 6

%e 6 7 6 6 2 3 3 7 5 5 4 6 3 7 6 5 7 3 0 4 0

%e 3 7 6 6 2 0 6 4 6 6 2 0 6 7 3 6 7 6 6 4 6

%e Vertex neighbors:

%e 0 -> 1 2 4

%e 1 -> 0 3 5

%e 2 -> 0 3 6

%e 3 -> 1 2 7

%e 4 -> 0 5 6

%e 5 -> 1 4 7

%e 6 -> 2 4 7

%e 7 -> 3 5 6

%Y Cf. A223331.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 19 2013

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)