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A183113
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Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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).
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REFERENCES
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Uri Levy, "The Magnetic Tower of Hanoi", Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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