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.

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

Offset: 0

Author

Uri Levy, Dec 25 2010

Keywords

• nonn
• easy

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.
• Web applet to play The Magnetic Tower of Hanoi
• Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-6).

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.

Programs

Mathematica

LinearRecurrence[{3,1,-1,-6},{0,1,3,9,25},30] (* Harvey P. Dale, Apr 30 2018 *)

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

Extensions

More terms from Harvey P. Dale, Apr 30 2018