A183122 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.

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

Offset: 0

Author

Uri Levy, Jan 07 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. 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).

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225
• U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arxiv:1011.3843
• U. Levy, to play The Magnetic Tower of Hanoi, web applet
• Index entries for linear recurrences with constant coefficients, signature (3, 1, -3).

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 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[{0,1,3,7},LinearRecurrence[{3,1,-3},{19,53,153},30]] (* Harvey P. Dale, Dec 08 2014 *)

Formula

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

Extensions

More terms from Harvey P. Dale, Dec 08 2014