A183116 Magnetic Tower of Hanoi, total number of moves, optimally solving the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 4, 11, 30, 85, 244, 715, 2118, 6309, 18860, 56475, 169262, 507541, 1522244, 4566155, 13697590, 41091429, 123272252, 369813659, 1109436254

Offset: 0

Author

Uri Levy, Dec 31 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 ; NEUTRAL] or [NEUTRAL ; 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 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*(7/11)*3^N ~ 0.5*0.636*3^(N). Series designation: S636(N).

References

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

Links

• Uri Levy, The Magnetic Tower of Hanoi, arxiv:1003.0225 [math.CO], 2010.
• Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
• Uri Levy, to play The Magnetic Tower of Hanoi, Web Applet. [Broken link]

Crossrefs

A183115 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 A183115 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.

A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

Programs

Mathematica

L1 = Root[-2 - # + #^3&, 1];
L2 = Root[-2 - # + #^3&, 3];
L3 = Root[-2 - # + #^3&, 2];
AP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 1];
BP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 3];
CP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 2];
(* b = A183115 *) b[0] = 0; b[n_] := (7/11) 3^(n-1) + AP (L1+1) L1^(n-1) + BP (L2+1) L2^(n-1) + CP (L3+1) L3^(n-1) // Round;
Array[b, 21, 0] // Accumulate (* Jean-François Alcover, Jan 30 2019 *)

Formula

G.f. appears to be (-4*x^3-3*x^2+1)/(-6*x^5+5*x^4+2*x^3+2*x^2-4*x+1).

Recurrence Relations (a(n)=S636(n) as in referenced paper):

S636(n) = S636(n-1) + 2*S909(n-2) + 3^(n-2) + 2 ; n >= 2 ; S909(0) = 0

Note: S909(n-2) refers to the integer sequence described by A183112.

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 n > 0:

S636(n) = (7/22)*3^n + AS*(λ1 + 1)*λ1^(n-1) + BS*(λ2 + 1)*λ2^(n-1) + CS*(λ3 + 1)*λ3^(n-1) - (3/2)