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A183113 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; BLUE] pre-colored puzzle. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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

LINKS

Table of n, a(n) for n=0..28.

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).

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

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 *)

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.

Sequence in context: A229188 A091486 A056779 * A102877 A122983 A005355

Adjacent sequences:  A183110 A183111 A183112 * A183114 A183115 A183116

KEYWORD

nonn

AUTHOR

Uri Levy, Dec 28 2010

EXTENSIONS

More terms from Harvey P. Dale, May 11 2011.

STATUS

approved

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)