A183115 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 3, 7, 19, 55, 159, 471, 1403, 4191, 12551, 37615, 112787, 338279, 1014703, 3043911, 9131435, 27393839, 82180823, 246541407, 739622595, 2218865335, 6656592255, 19969771063, 59909304539, 179727900415, 539183681191, 1617551013071, 4852652992755, 14557958907655, 43673876615503

Offset

0,3

Comments

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configurations). Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Disk numbering is from largest disk (k = 1) to smallest disk (k = N)

The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.

Number of moves of disk k, for large k, is close to (7/11)*3^(k-1) ~ 0.636*3^(k-1). Series designation: P636(k).

References

Uri Levy, "The Magnetic Tower of Hanoi", Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp. 173.

Links

Table of n, a(n) for n=0..30.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.

Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.

Web applet to play The Magnetic Tower of Hanoi [Broken link]

Formula

Recurrence Relations (a(n)=P636(n) as in referenced paper):

P636(n) = P636(n-1) + 2*P909(n-2) + 2*3^(n-3) ; n >= 3

Note: P909(n-2) refers to the integer sequence described by A183111.

Closed-Form Expression:

Define:

λ1 = [1+sqrt(26/27)]^(1/3) + [1-sqrt(26/27)]^(1/3)

λ2 = -0.5* λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}

λ3 = -0.5* λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}

AP = [(1/11)* λ2* λ3 - (3/11)*(λ2 + λ3) + (9/11)]/[( λ2 - λ1)*( λ3 - λ1)]

BP = [(1/11)* λ1* λ3 - (3/11)*(λ1 + λ3) + (9/11)]/[( λ1 - λ2)*( λ3 - λ2)]

CP = [(1/11)* λ1* λ2 - (3/11)*(λ1 + λ2) + (9/11)]/[( λ2 - λ3)*( λ1 - λ3)]

For n > 0:

P636(n) = (7/11)*3^(n-1) + AP*(λ1+1)*λ1^(n-1) + BP*( λ2+1)*λ2^(n-1) + CP*(λ3+1)* λ3^(n-1)

Empirical G.f.: x*(1-3*x^2-4*x^3)/((1-3*x)*(1-x^2-2*x^3)). [Colin Barker, Jan 12 2012]

Mathematica

L1 = Root[-2 - # + #^3&, 1];
L2 = Root[-2 - # + #^3&, 3];
L3 = Root[-2 - # + #^3&, 2];
AP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 1];
BP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 3];
CP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 2];
a[0] = 0;
a[n_] := (7/11) 3^(n-1) + AP (L1+1) L1^(n-1) + BP (L2+1) L2^(n-1) + CP (L3+1) L3^(n-1);
Table[a[n] // Round, {n, 0, 30}] (* Jean-François Alcover, Dec 03 2018 *)

Crossrefs

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.

A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

Sequence in context: A351633 A115760 A175533 * A183120 A100702 A367484

Adjacent sequences: A183112 A183113 A183114 * A183116 A183117 A183118

Keyword

nonn

Author

Uri Levy, Dec 31 2010

Extensions

More terms from Jean-François Alcover, Dec 03 2018

Status

approved