

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

Table of n, a(n) for n=0..28.
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.
Web applet to play The Magnetic Tower of Hanoi
Index entries for linear recurrences with constant coefficients, signature (3,1,1,6).


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)
33*a(n) = 10*3^n 3*( A052947(n2) A052947(n1) 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 kth disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] precolored Magnetic Tower of Hanoi.
Cf. A183111  A183125.
Sequence in context: A004665 A196431 A244826 * A132835 A191354 A001189
Adjacent sequences: A183108 A183109 A183110 * A183112 A183113 A183114


KEYWORD

nonn,easy


AUTHOR

Uri Levy, Dec 25 2010


EXTENSIONS

More terms from Harvey P. Dale, Apr 30 2018


STATUS

approved



