

A183111


Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] precolored 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
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OFFSET

0,3


COMMENTS

A. The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is precolored. Precoloring 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 precoloring 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 abovelisted "original" sequence generates a "partialsums" 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^(k1) ~ 0.909*3^(k1). 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.


FORMULA

G.f.: x*(1+4*x^3+x^2) / ( (3*x1)*(2*x^3+x^21) ).
Recurrence Relations (a(n)=P909(n) as in referenced paper):
a(n) = a(n2) + a(n3) + 2*3^(n2) + 2*3^(n4) ; n >= 4
ClosedForm Expression:
Define:
λ1 = [1+sqrt(26/27)]^(1/3) + [1sqrt(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^(n1) + AP* λ1^(n1) + BP* λ2^(n1) + CP* λ3^(n1)


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 kth disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] precolored Magnetic Tower of Hanoi.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



