A183120 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 [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.

0, 1, 3, 7, 19, 55, 159, 471, 1403, 4199, 12583, 37735, 113187, 339543, 1018607, 3055799, 9167371, 27502087, 82506231, 247518663, 742555955, 2227667831, 6683003455, 20049010327, 60147030939, 180441092775, 541323278279, 1623969834791, 4871909504323

Offset: 0

Author

Uri Levy, Jan 05 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 [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the presented sequence is NOT optimal. The particular "64" algorithm solving the puzzle at hand is not explicitly presented in any of the referenced papers. The series and its properties are listed in the paper referenced by link 2 listed below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration see A183115 and A183116. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

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 (23/36)*3^(k-1) ~ 0.64*3^(k-1). Series designation: P64(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

Cf. A100702 - is a sequence also describing the number of moves of disk number k, generated by another algorithm, designated "67", yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle at hand. Recurrence relations for this sequence is a(k) = 3*a(k-1) - 2 and the closed-form expression is (2/3)*3^(k-1)+1. Large k limit is clearly (2/3)*3^(k-1) =~ 0.67*3^(k-1), and sequence designation is thus P67(k). The (non-optimal) "67" algorithm solving the Magnetic Tower of Hanoi with the given pre-coloring configuration yielding the P67(k) sequence (given by A100702) is explicitly described and discussed in the paper referenced in link 1 above.

Cf. 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.

Programs

Mathematica

nxt[{a_,b_}]:=Module[{c=3b-2(a+1)},{a+1,If[EvenQ[a+1],c+6,c+8]}]; Join[ {0,1,3,7},Transpose[NestList[nxt,{4,19},25]][[2]]] (* or *) Join[ {0,1,3},LinearRecurrence[{4,-2,-4,3},{7,19,55,159},40]] (* Harvey P. Dale, May 04 2012 *)

Formula

G.f.: x*(3*x^2-x^3-2*x^4+4*x^5-1+x) / ((1+x)*(3*x-1)*(x-1)^2), equivalent to a(n) = 23*3^n/108+n-2-(-1)^n/4 for n>2.

(a(n) = P64(n) as in referenced paper):

a(n) = 3*a(n-1) - 2*n + 6; n even; n >= 4

a(n) = 3*a(n-1) - 2*n + 8; n odd; n >= 5

a(n) = a(n-1) + 2* P75(n-3) + 10*3^(n-4); n >= 4

P75(n) refers to the integer sequence described by A122983. See also A183119.

a(n) = (23/36)*3^(n-1) + n - 9/4; n even; n >= 4

a(n) = (23/36)*3^(n-1) + n - 7/4; n odd; n >= 3

a(n) = 4*a(n-1)- 2*a(n-2)-4*a(n-3)+3*a(n-4). [Harvey P. Dale, May 04 2012]

Extensions

More terms from Harvey P. Dale, May 04 2012