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
Uri Levy, "The Magnetic Tower of Hanoi", Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
LINKS
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.
KEYWORD
nonn
AUTHOR
Uri Levy, Dec 28 2010
EXTENSIONS
More terms from Harvey P. Dale, May 11 2011
STATUS
approved