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A183115
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Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.
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4
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0, 1, 3, 7, 19, 55, 159, 471, 1403, 4191, 12551, 37615, 112787, 338279, 1014703, 3043911, 9131435, 27393839, 82180823, 246541407, 739622595, 2218865335, 6656592255, 19969771063, 59909304539, 179727900415, 539183681191, 1617551013071, 4852652992755, 14557958907655, 43673876615503
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OFFSET
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0,3
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COMMENTS
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The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; 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 configurations). 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 (7/11)*3^(k-1) ~ 0.636*3^(k-1). Series designation: P636(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)=P636(n) as in referenced paper):
P636(n) = P636(n-1) + 2*P909(n-2) + 2*3^(n-3) ; n >= 3
Note: P909(n-2) refers to the integer sequence described by A183111.
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 > 0:
P636(n) = (7/11)*3^(n-1) + AP*(λ1+1)*λ1^(n-1) + BP*( λ2+1)*λ2^(n-1) + CP*(λ3+1)* λ3^(n-1)
Empirical G.f.: x*(1-3*x^2-4*x^3)/((1-3*x)*(1-x^2-2*x^3)). [Colin Barker, Jan 12 2012]
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MATHEMATICA
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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_] := (7/11) 3^(n-1) + AP (L1+1) L1^(n-1) + BP (L2+1) L2^(n-1) + CP (L3+1) L3^(n-1);
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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 puzzle.
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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