OFFSET
0,3
COMMENTS
A. The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [NEUTRAL ; NEUTRAL ; NEUTRAL], given in [Source ; Intermediate ; Destination] order. Thus, the tower in this case is "natural" or "free". The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the "free" Magnetic Tower of Hanoi puzzle). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
B. Number of moves to solve the given puzzle, for large N, is close to 0.5*(20/33)*3^N ~ 0.5*0.606*3^(N). Series designation: S606(N).
C. The large N ratio of number of moves to solve the [NEUTRAL ; NEUTRAL ; NEUTRAL] puzzle to the number of moves to solve the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] puzzle is 20/33 or about 60.6% (see link 2).
REFERENCES
"The Magnetic Tower of Hanoi", Uri Levy, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
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, The Magnetic Tower of Hanoi, 2009.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6).
FORMULA
G.f. x*(1 + x)*(1 - x - 2*x^2 - 2*x^3)/((1 - x)*(1 - 3*x)*(1 - x^2 - 2*x^3)).
Recurrence Relations (a(n)=S606(n) as in referenced paper):
a(n) = a(n-1) + a(n-2) + a(n-2) + 3^(n-2) + 2, for n >= 2; a(0) = 0, a(1) = 1.
Closed-Form Expression: Let
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)]
Then, for n > 0:
a(n) = (10/33)*3^n + 0.5*AS*[(λ1 + 1)^2]*λ1^(n-1) + 0.5*BS*[(λ2 + 1)^2]*λ2^(n-1) + 0.5*CS*[(λ3 + 1)^2]*λ3^(n-1) - 2.
MATHEMATICA
Join[{0}, LinearRecurrence[{4, -2, -2, -5, 6}, {1, 4, 11, 30, 83}, 20]] (* Jean-François Alcover, Jan 28 2019 *)
CROSSREFS
A183117 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 A183117 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.
KEYWORD
nonn,easy
AUTHOR
Uri Levy, Jan 01 2011
EXTENSIONS
a(21) onward from Andrew Howroyd, Nov 08 2025
STATUS
approved