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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 [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.

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

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

D.   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

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225

Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843

Uri Levy, to play The Magnetic Tower of Hanoi, web applet

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(0)=0, a(1)=1, a(2)=3, a(3)=7, a(4)=19, a(5)=55, a(6)=159, a(n)= 4*a(n-1)- 2*a(n-2)-4*a(n-3)+3*a(n-4) [From Harvey P. Dale, May 04 2012]

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 *)

CROSSREFS

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.

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.

A183111 - A183125.

Sequence in context: A175533 A115760 A183115 * A100702 A224031 A147586

Adjacent sequences:  A183117 A183118 A183119 * A183121 A183122 A183123

KEYWORD

nonn

AUTHOR

Uri Levy, Jan 05 2011

EXTENSIONS

More terms from Harvey P. Dale, May 04 2012

STATUS

approved

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Last modified August 13 08:34 EDT 2020. Contains 336442 sequences. (Running on oeis4.)