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A183117 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

%I A183117 #34 Dec 04 2018 04:09:10
%S A183117 0,1,3,7,19,53,153,451,1339,3997,11961,35835,107435,322197,966425,
%T A183117 2899027,8696699,26089517,78267673,234801675,704402987,2113205861,
%U A183117 6339612857,19018831395,57056483259,171169433149,513508274169,1540524784027,4621574293547,13864722791605,41594168239321
%N A183117 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.
%C A183117 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. Thus, the tower in this case is "natural" or "free". 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.
%C A183117 Disk numbering is from largest disk (k = 1) to smallest disk (k = N).
%C A183117 The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
%C A183117 Number of moves of disk k, for large k, is close to (20/33)*3^(k-1) ~ 0.606*3^(k-1). Series designation: P606(k).
%D A183117 Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp. 173.
%H A183117 Uri Levy, <a href="http://arxiv.org/abs/1003.0225">The Magnetic Tower of Hanoi</a>, arXiv:1003.0225 [math.CO], 2010.
%H A183117 Uri Levy, <a href="http://arxiv.org/abs/1011.3843">Magnetic Towers of Hanoi and their Optimal Solutions</a>, arXiv:1011.3843 [math.CO], 2010.
%H A183117 Web applet <a href="http://www.weizmann.ac.il/zemed/davidson_online/mtoh/MTOHeng.html">to play The Magnetic Tower of Hanoi</a> [Broken link]
%F A183117 G.f. appears to be -x*(1+x)*(2*x^3+2*x^2+x-1)/((3*x-1)*(2*x^3+x^2-1)) with a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 6*a(n-4) for n > 5. - _Joerg Arndt_, Jan 03 2011
%F A183117 Recurrence Relations (a(n)=P606(n) as in referenced paper):
%F A183117 P606(n) = P636(n-1) + P636(n-2) + P909(n-2) + 2*3^(n-3) ; n >= 3.
%F A183117 Note:  P636(n) and P909(n) refer to the integer sequences described by A183115 and A183111 respectively.
%F A183117 Closed-Form Expression:
%F A183117 Define:
%F A183117 λ1 = (1+sqrt(26/27))^(1/3) + (1-sqrt(26/27))^(1/3)
%F A183117 λ2 = -0.5*λ1 + 0.5*i*((sqrt(27) + sqrt(26))^(1/3) - (sqrt(27) - sqrt(26))^(1/3))
%F A183117 λ3 = -0.5*λ1 - 0.5*i*((sqrt(27) + sqrt(26))^(1/3) - (sqrt(27) - sqrt(26))^(1/3))
%F A183117 AP = ((1/11)*λ2*λ3 - (3/11)*(λ2 + λ3) + (9/11))/((λ2 - λ1)*(λ3 - λ1))
%F A183117 BP = ((1/11)*λ1*λ3 - (3/11)*(λ1 + λ3) + (9/11))/((λ1 - λ2)*(λ3 - λ2))
%F A183117 CP = ((1/11)*λ1*λ2 - (3/11)*(λ1 + λ2) + (9/11))/((λ2 - λ3)*(λ1 - λ3))
%F A183117 For n > 1:
%F A183117 P606(n) = (20/33)*3^(n-1) + 0.5*AP*((λ1+1)^2)*λ1^(n-1) + 0.5*BP*((λ2+1)^2)*λ2^(n-1) + 0.5*CP*(λ3+1)^2)*λ3^(n-1).
%t A183117 L1 = Root[-2 - # + #^3&, 1];
%t A183117 L2 = Root[-2 - # + #^3&, 3];
%t A183117 L3 = Root[-2 - # + #^3&, 2];
%t A183117 AP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 1];
%t A183117 BP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 3];
%t A183117 CP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 2];
%t A183117 a[0] = 0;
%t A183117 a[n_] := (1/2) AP (L1+1)^2 L1^(n-1) + (1/2) BP (L2+1)^2 L2^(n-1) + (1/2) CP (L3+1)^2 L3^(n-1) + (20 3^(n-1))/33;
%t A183117 Table[a[n] // Round, {n, 0, 30}] (* _Jean-François Alcover_, Dec 03 2018 *)
%Y A183117 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.
%Y A183117 A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.
%K A183117 nonn
%O A183117 0,3
%A A183117 _Uri Levy_, Dec 31 2010