A183113 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 3, 7, 21, 61, 179, 535, 1597, 4781, 14331, 42967, 128869, 386557, 1159587, 3478647, 10435757, 31306989, 93920555, 281761015, 845282069, 2535844733, 7607531923, 22822592343, 68467771805, 205403307437, 616209910235, 1848629712279, 5545889108805

Offset: 0

Author

Uri Levy, Dec 28 2010

Keywords

• nonn

Comments

A. The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; 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 configuration). Optimal solutions are discussed and their optimality is proved in link 2 listed below.

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

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

D. Number of moves of disk k, for large k, is close to (8/11)*3^(k-1) ~ 0.727*3^(k-1). Series designation: P727(k).

References

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

Links

• 1. The Magnetic Tower of Hanoi, Uri Levy
• 2. Magnetic Towers of Hanoi and their Optimal Solutions, Uri Levy
• 3. Web applet to play The Magnetic Tower of Hanoi
• Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-6).

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.

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

Programs

Mathematica

Join[{0},LinearRecurrence[{3,1,-1,-6},{1,3,7,21},40]] (* or *) CoefficientList[ Series[ x(1-2x)(1+x)^2/((1-3x)(1-x^2-2x^3)),{x,0,40}],x] (* Harvey P. Dale, May 11 2011 *)

Formula

Recurrence Relations (a(n)=P727(n) as in referenced paper):

P727(k) = P727(k-2) + 2*P727(k-3) + 4*3^(k-3) + 4*3^(k-4) ; k >= 4

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 any k > 0:

P727(n) = (8/11)*3^(n-1) + AP* λ1^n + BP* λ2^n + CP* λ3^n.

G.f.: x*(1-2*x)*(1+x)^2/((1-3*x)*(1-x^2-2*x^3)); a(n) = 3*a(n-1)+a(n-2)-a(n-3)-6*a(n-4) with n>4. - Bruno Berselli, Dec 29 2010

Extensions

More terms from Harvey P. Dale, May 11 2011