A183123 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

0, 1, 4, 11, 30, 83, 236, 691, 2050, 6123, 18336, 54971, 164870, 494563, 1483636, 4450851, 13352490, 40057403, 120172136, 360516331, 1081548910, 3244646643, 9733939836, 29201819411, 87605458130, 262816374283, 788449122736, 2365347368091, 7096042104150

Offset: 0

Author

Uri Levy, Jan 07 2011

Keywords

• nonn
• easy

Comments

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. The solution algorithm producing the presented sequence is NOT optimal. The particular "62" algorithm solving the puzzle at hand is presented and discussed in the paper referenced by link 1 below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration (the "natural" or "free" Magnetic Tower) see A183117 and A183118. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Large N limit of the sequence is 0.5*(67/108)*3^N ~ 0.5*0.62*3^N. Series designation: S62(n).

References

• U. 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 [math.CO], 2010.
• U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arxiv:1011.3843 [math.CO], 2010.
• U. Levy, to play The Magnetic Tower of Hanoi, web applet.
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Crossrefs

Cf. A183122 - "Magnetic Tower of Hanoi, number of moves of disk number k, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle" is an "original" sequence describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "62" algorithm mentioned above.

Cf. 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

LinearRecurrence[{4,-2,-4,3},{0,1,4,11,30,83,236},40] (* Harvey P. Dale, Jun 07 2015 *)

PARI

concat(0, Vec(x*(4*x^5+2*x^4+2*x^3+3*x^2-1)/((x-1)^2*(x+1)*(3*x-1)) + O(x^100))) \\ Colin Barker, Sep 18 2014

Formula

a(n)=+4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4) for n>6.

(a(n) = S62(n) as in referenced paper):

S62(n) = 3*S62(n-1) - 5*n + 17; n even; n >= 4

S62(n) = 3*S62(n-1) - 5*n + 13; n odd; n >= 5

S62(n) = S67(n-1) + S67(n-2) + S75(n-3) + 4*3^(n-3) + 2; n >= 3

S67(n) and S75(n) refer to the integer sequences described by A104743 and A183119 respectively.

S62(n) = 0.5*(67/108)*3^n + 2.5*n - 41/8; n even; n >= 4

S62(n) = 0.5*(67/108)*3^n + 2.5*n - 39/8; n odd; n >= 3.

a(n) = -5-(-1)^n/8+(67*3^(-3+n))/8+(5*n)/2 for n>2. - Colin Barker, Sep 18 2014

G.f.: x*(4*x^5+2*x^4+2*x^3+3*x^2-1) / ((x-1)^2*(x+1)*(3*x-1)). - Colin Barker, Sep 18 2014

Extensions

More terms and correction to recurrence by Colin Barker, Sep 18 2014