A183118 Magnetic Tower of Hanoi, total number of moves, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

0, 1, 4, 11, 30, 83, 236, 687, 2026, 6023, 17984, 53819, 161254, 483451, 1449876, 4348903, 13045602, 39135119, 117402792, 352204467, 1056607454

Offset: 0

Author

Uri Levy, Jan 01 2011

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 [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 "free" Magnetic Tower of Hanoi puzzle). Optimal solutions are discussed and their optimality is proved in link 2 listed below.

B. Number of moves to solve the given puzzle, for large N, is close to 0.5*(20/33)*3^N ~ 0.5*0.606*3^(N). Series designation: S606(N).

C. The large N ratio of number of moves to solve the [NEUTRAL ; NEUTRAL ; NEUTRAL] puzzle to the number of moves to solve the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] puzzle is 20/33 or about 60.6% (see link 2).

References

• "The Magnetic Tower of Hanoi", Uri Levy, 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
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6).

Crossrefs

A183117 is an "original" sequence describing the number of moves of disk number k, optimally solving the pre-colored puzzle at hand. The integer sequence listed above is the partial sums of the A183117 original sequence.

A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves 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

Join[{0}, LinearRecurrence[{4, -2, -2, -5, 6}, {1, 4, 11, 30, 83}, 20]] (* Jean-François Alcover, Jan 28 2019 *)

Formula

G.f. x*(-2*x^4-4*x^3-3*x^2+1)/(-6*x^5+5*x^4+2*x^3+2*x^2-4*x+1).

Recurrence Relations (a(n)=S606(n) as in referenced paper):

S606(n) = S636(n-1)+ S636(n-2)+ S909(n-2)+ 3^(n-2)+ 2; n >= 2; S909(0) = 0; S636(0) = 0

Note: S636(n) and S909(n) are sequences A183116 and A183112 respectively.

Closed-Form Expression: Let

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)}

AS = [(7/11)* λ2* λ3 - (10/11)*(λ2 + λ3) + (19/11)]/[(λ2 - λ1)*( λ3 - λ1)]

BS = [(7/11)* λ1* λ3 - (10/11)*(λ1 + λ3) + (19/11)]/[(λ1 - λ2)*( λ3 - λ2)]

CS = [(7/11)* λ1* λ2 - (10/11)*(λ1 + λ2) + (19/11)]/[(λ2 - λ3)*( λ1 - λ3)]

Then, for n > 0:

S606(n) = (10/33)*3^n + 0.5*AS*[(λ1 + 1)^2]*λ1^(n-1) + 0.5*BS*[(λ2 + 1)^2]*λ2^(n-1) + 0.5*CS*[(λ3 + 1)^2]*λ3^(n-1) - 2