A183114 Magnetic Tower of Hanoi, total number of moves, optimally solving the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 4, 11, 32, 93, 272, 807, 2404, 7185, 21516, 64483, 193352, 579909, 1739496, 5218143, 15653900, 46960889, 140881444, 422642459, 1267924528, 3803769261, 11411301184, 34233893527, 102701665332, 308104972769, 924314883004, 2772944595283, 8318833704088, 24956500987925

Offset: 0

Author

Uri Levy, Dec 29 2010

Keywords

• nonn

Comments

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; 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 configuration). Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Number of moves to solve the given puzzle, for large N, is close to 0.5*(8/11)*3^N ~ 0.5*0.727*3^(N). Series designation: S727(N).

References

• U. Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2030
• U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
• U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
• Web applet to play The Magnetic Tower of Hanoi [Broken link]
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6).

Crossrefs

A183113 is an "original" sequence describing the number of moves of disk number k, optimally solving the pre-colored puzzle at hand. The integer sequence listed above is the partial sums of the A183113 original sequence.

A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.

Cf. A183111 - A183125.

Programs

Magma

I:=[0,1,4,11,32]; [n le 5 select I[n] else 4*Self(n-1)-2*Self(n-2)-2*Self(n-3)-5*Self(n-4)+6*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 04 2018

Maple

seq(coeff(series(x*(2*x-1)*(1+x)^2/((x-1)*(3*x-1)*(2*x^3+x^2-1)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Dec 04 2018

Mathematica

LinearRecurrence[{4, -2, -2, -5, 6}, {0, 1, 4, 11, 32}, 30] (* Jean-François Alcover, Dec 04 2018 *)
CoefficientList[Series[x (2 x - 1) (1 + x)^2 / ((x - 1) (3 x - 1) (2 x^3 + x^2 - 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Dec 04 2018 *)

Formula

G.f.: x*(2*x-1)*(1+x)^2 / ( (x-1)*(3*x-1)*(2*x^3+x^2-1) ).

Recurrence Relations (a(n)=S727(n) as in referenced paper):

a(N) = a(N-2) + 2*a(N-3) + 8*3^(N-3) + 2 ; N ≥ 3 ; S727(0) = 0

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)}

AS = [(7/11)* λ2* λ3 - (10/11)*(λ2 + λ3) + (19/11)]/[( λ2 - λ1)*( λ3 - λ1)]

BS = [(7/11)* λ1* λ3 - (10/11)*(λ1 + λ3) + (19/11)]/[( λ1 - λ2)*( λ3 - λ2)]

CS = [(7/11)* λ1* λ2 - (10/11)*(λ1 + λ2) + (19/11)]/[( λ2 - λ3)*( λ1 - λ3)]

For any N > 0:

a(N) = (4/11)*3^N + AS* λ1^N + BS* λ2^N + CS* λ3^N - 1

a(n) = 4*a(n-1)-2*a(n-2)-2*a(n-3)-5*a(n-4)+6*a(n-5). - Vincenzo Librandi, Dec 04 2018

Extensions

More terms from Jean-François Alcover, Dec 04 2018