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A183122
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Magnetic Tower of Hanoi, number of moves of disk number k, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.
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3
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0, 1, 3, 7, 19, 53, 153, 455, 1359, 4073, 12213, 36635, 109899, 329693, 989073, 2967215, 8901639, 26704913, 80114733, 240344195, 721032579, 2163097733, 6489293193, 19467879575, 58403638719, 175210916153, 525632748453, 1576898245355, 4730694736059
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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 [NEUTRAL ; NEUTRAL ; NEUTRAL], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the presented sequence is NOT optimal. The particular "62" algorithm solving the puzzle at hand is presented and discussed in the paper referenced by link 1 below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration (the "natural" or "free" Magnetic Tower) see A183117 and A183118. 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 (67/108)*3^(k-1) ~ 0.62*3^(k-1). Series designation: P62(k).
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REFERENCES
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U. 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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a(n)=+3*a(n-1)+a(n-2)-3*a(n-3) for n>6.
g.f.: x+ 3*x^2 +7*x^3 -x^4*(-19+4*x+25*x^2)/ ((x-1)(3*x-1)(1+x)).
(a(n) = P62(n) as in referenced paper):
a(n) = 3*a(n-1) - 6; n even; n >= 6
a(n) = 3*a(n-1) - 4; n odd; n >= 5
a(n) = P67(n-1) + P67(n-2) + P75(n-3) + 8*3^(n-4) ; n >= 4
P75(n) and P67(n) refer to the integer sequences described by A122983 and A100702 respectively. See also A183119.
a(n) = (67/108)*3^(n-1) + 9/4; n even; n >= 4
a(n) = (67/108)*3^(n-1) + 11/4; n odd; n >= 5
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MATHEMATICA
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Join[{0, 1, 3, 7}, LinearRecurrence[{3, 1, -3}, {19, 53, 153}, 30]] (* Harvey P. Dale, Dec 08 2014 *)
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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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