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A183111 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle. 20
0, 1, 3, 9, 25, 75, 223, 665, 1993, 5971, 17903, 53697, 161065, 483163, 1449439, 4348233, 13044585, 39133571, 117400431, 352200881, 1056601993, 3169805003, 9509413535, 28528238329, 85584711561, 256754129459, 770262380399, 2310787129121, 6932361368937 (list; graph; refs; listen; history; text; internal format)
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 ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; 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.
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. The number of moves of disk k, for large k, is close to (10/11)*3^(k-1) ~ 0.909*3^(k-1). Series designation: P909(k).
LINKS
Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173; arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
FORMULA
G.f.: -x*(-1+4*x^3+x^2) / ( (3*x-1)*(2*x^3+x^2-1) ).
Recurrence Relations (a(n)=P909(n) as in referenced paper):
a(n) = a(n-2) + a(n-3) + 2*3^(n-2) + 2*3^(n-4) ; n >= 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 n > 0:
a(n) = (10/11)*3^(n-1) + AP* λ1^(n-1) + BP* λ2^(n-1) + CP* λ3^(n-1)
33*a(n) = 10*3^n -3*( A052947(n-2) -A052947(n-1) -4*A052947(n) ). - R. J. Mathar, Feb 05 2020
MATHEMATICA
LinearRecurrence[{3, 1, -1, -6}, {0, 1, 3, 9, 25}, 30] (* Harvey P. Dale, Apr 30 2018 *)
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.
Sequence in context: A004665 A196431 A244826 * A132835 A191354 A001189
KEYWORD
nonn,easy
AUTHOR
Uri Levy, Dec 25 2010
EXTENSIONS
More terms from Harvey P. Dale, Apr 30 2018
STATUS
approved

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Last modified April 23 18:16 EDT 2024. Contains 371916 sequences. (Running on oeis4.)