A183117 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

0, 1, 3, 7, 19, 53, 153, 451, 1339, 3997, 11961, 35835, 107435, 322197, 966425, 2899027, 8696699, 26089517, 78267673, 234801675, 704402987, 2113205861, 6339612857, 19018831395, 57056483259, 171169433149, 513508274169, 1540524784027, 4621574293547, 13864722791605, 41594168239321

Offset: 0

Author

Uri Levy, Dec 31 2010

Keywords

• nonn

Comments

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [NEUTRAL ; NEUTRAL ; NEUTRAL], given in [Source ; Intermediate ; Destination] order. Thus, the tower in this case is "natural" or "free". 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 configuration). 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 (20/33)*3^(k-1) ~ 0.606*3^(k-1). Series designation: P606(k).

References

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

Links

• 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]

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.

Programs

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_] := (1/2) AP (L1+1)^2 L1^(n-1) + (1/2) BP (L2+1)^2 L2^(n-1) + (1/2) CP (L3+1)^2 L3^(n-1) + (20 3^(n-1))/33;
Table[a[n] // Round, {n, 0, 30}] (* Jean-François Alcover, Dec 03 2018 *)

Formula

G.f. appears to be -x*(1+x)*(2*x^3+2*x^2+x-1)/((3*x-1)*(2*x^3+x^2-1)) with a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 6*a(n-4) for n > 5. - Joerg Arndt, Jan 03 2011

Recurrence Relations (a(n)=P606(n) as in referenced paper):

P606(n) = P636(n-1) + P636(n-2) + P909(n-2) + 2*3^(n-3) ; n >= 3.

Note: P636(n) and P909(n) refer to the integer sequences described by A183115 and A183111 respectively.

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 > 1:

P606(n) = (20/33)*3^(n-1) + 0.5*AP*((λ1+1)^2)*λ1^(n-1) + 0.5*BP*((λ2+1)^2)*λ2^(n-1) + 0.5*CP*(λ3+1)^2)*λ3^(n-1).