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A183112 Magnetic Tower of Hanoi, total number of moves, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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

LINKS

Table of n, a(n) for n=0..28.

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).

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

MATHEMATICA

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

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.

Sequence in context: A159036 A058693 A027076 * A266429 A105693 A215404

Adjacent sequences:  A183109 A183110 A183111 * A183113 A183114 A183115

KEYWORD

nonn

AUTHOR

Uri Levy, Dec 25 2010

STATUS

approved

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Last modified December 4 12:21 EST 2020. Contains 338923 sequences. (Running on oeis4.)