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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

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

U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arxiv:1011.3834

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*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)

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.

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

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Last modified December 6 18:03 EST 2019. Contains 329809 sequences. (Running on oeis4.)