A183124 Magnetic Tower of Hanoi, number of moves of disk number n, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

0, 1, 3, 7, 19, 53, 153, 451, 1339, 4001, 11981, 35919, 107727, 323149, 969409, 2908187, 8724515, 26173497, 78520437, 235561255, 706683703

Offset: 0

Author

Uri Levy, Jan 08 2011

Keywords

• nonn
• easy

Comments

The Magnetic Tower of Hanoi puzzle is described in the preprint of March 2010. 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 "61" algorithm solving the puzzle at hand is not explicitly presented in any of the referenced papers. 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 the preprint of Nov 2010.

Disk numbering is from largest disk (k = 1) to smallest disk (k = N).

The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.

Number of moves of disk k, for large k, is close to (197/324)*3^(k-1) ~ 0.61*3^(k-1). Series designation: P61(k).

References

• Uri Levy, The Magnetic Tower of Hanoi, 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.
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Crossrefs

A183122 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.

Cf. A183111-A183125.

Formula

G.f.: (-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(-3*x^4 +4*x^3 +2*x^2 -4*x +1)

a(n)=+4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4), n>=9.

(a(n) = P61(n) as in referenced paper):

a(n) = 3*a(n-1) - 4*n + 18 ; n even ; n >= 5

a(n) = 3*a(n-1) - 4*n + 20 ; n odd ; n >= 6

a(n) = P64(n-1) + P64(n-2) + P75(n-3) + 8*3^(n-4) ; n >= 4

P75(n) and P64(n) refer to the integer sequences described by A122983 and A183120 respectively. See also A183119.

a(n) = (197/324)*3^(n-1) + 2*n - 27/4; n even; n >= 6

a(n) = (197/324)*3^(n-1) + 2*n - 25/4; n odd; n >= 5