A183112 Magnetic Tower of Hanoi, total number of moves, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle.

0, 1, 4, 13, 38, 113, 336, 1001, 2994, 8965, 26868, 80565, 241630, 724793, 2174232, 6522465, 19567050, 58700621, 176101052, 528301933, 1584903926, 4754708929, 14264122464, 42792360793, 128377072354, 385131201813, 1155393582212, 3466180711333, 10398542080270

Offset: 0

Author

Uri Levy, Dec 25 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 ; 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.

Number of moves to solve the given puzzle, for large N, is close to 0.5*(10/11)*3^N ~ 0.5*0.909*3^(N). Series designation: S909(N).

References

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

Links

• U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225
• Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843
• Web applet to play The Magnetic Tower of Hanoi
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6).

Crossrefs

A183111 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 A183111 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

Mathematica

LinearRecurrence[{4, -2, -2, -5, 6}, {0, 1, 4, 13, 38}, 21] (* Jean-François Alcover, Dec 14 2018 *)

Formula

Recurrence Relations (a(n)=S909(n) as in the referenced papers):

a(n) = a(n-2) + a(n-3) + 3^(n-1) + 3^(n-3) + 2; n >= 3 ; a(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) = (5/11)*3^n + AS* λ1^(n-1) + BS* λ2^(n-1) + CS* λ3^(n-1) - 1.

G.f.: x*(1-x^2-4*x^3)/((1-x)*(1-3*x)*(1-x^2-2*x^3)); a(n)=4*a(n-1)-2*a(n-2)-2*a(n-3)-5*a(n-4)+6*a(n-5) with n>5. - Bruno Berselli, Dec 29 2010