A183113 -id:A183113 - cp's OEIS frontend

A183119 Magnetic Tower of Hanoi, total number of moves generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 4, 11, 32, 93, 276, 823, 2464, 7385, 22148, 66435, 199296, 597877, 1793620, 5380847, 16142528, 48427569, 145282692, 435848059, 1307544160, 3922632461, 11767897364, 35303692071, 105911076192, 317733228553, 953199685636, 2859599056883, 8578797170624, 25736391511845

Offset: 0

Uri Levy, Jan 03 2011

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 presented sequence is NOT optimal. The particular "75" algorithm solving the puzzle at hand is presented in a paper referenced by link 1 listed below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration see A183113 and A183114. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Large N limit of the sequence is 0.5*(3/4)*3^N = 0.5*0.75*3^N. Series designation: S75(n).

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

Muniru A Asiru, Table of n, a(n) for n = 0..2070
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]
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

A122983 - "Binomial transform of aeration of A081294" is an "original" sequence (also) describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "75" algorithm mentioned above and presented in the paper referenced by link 1 above. The integer sequence listed above is the partial sums of the A122983 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.

[3^(n+1)/8+(n-1)/2+(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Dec 04 2018

seq(coeff(series(x*(3*x^2-1)/((1+x)*(3*x-1)*(x-1)^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 04 2018

LinearRecurrence[{4, -2, -4, 3}, {0, 1, 4, 11}, 30] (* Jean-François Alcover, Dec 04 2018 *)
Table[3^(n + 1) / 8 + (n - 1) / 2 + (-1)^n / 8, {n, 0, 30}] (* Vincenzo Librandi, Dec 04 2018 *)

a(n) = 3^(n+1)/8 + (n-1)/2 + (-1)^n/8 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(-1+3*x^2) / ( (1+x)*(3*x-1)*(x-1)^2 ).
(a(n)=S75(n) in referenced paper):
a(n) = 3*a(n-1) - n + 3; n even; n >= 2
a(n) = 3*a(n-1) - n + 2; n odd; n >= 1
a(n) = a(n-2) + 3^(n-1) + 1; n >= 2
a(n) = 3^(n+1)/8 + (n-1)/2 +(-1)^n/8.

More terms from Jean-François Alcover, Dec 04 2018

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

Original entry on oeis.org

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

Uri Levy, Dec 29 2010

nonn

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).

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

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).

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.

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

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

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 *)

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

More terms from Jean-François Alcover, Dec 04 2018

cp's OEIS Frontend

A183119 Magnetic Tower of Hanoi, total number of moves generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

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