cp's OEIS Frontend

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.

%I A183124 #25 May 06 2024 01:45:34
%S A183124 0,1,3,7,19,53,153,451,1339,4001,11981,35919,107727,323149,969409,
%T A183124 2908187,8724515,26173497,78520437,235561255,706683703
%N 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.
%C A183124 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.
%C A183124 Disk numbering is from largest disk (k = 1) to smallest disk (k = N).
%C A183124 The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
%C A183124 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).
%D A183124 Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
%H A183124 Uri Levy, <a href="http://arxiv.org/abs/1003.0225">The Magnetic Tower of Hanoi</a>, arXiv:1003.0225 [math.CO], 2010.
%H A183124 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 A183124 Uri Levy, <a href="http://www.weizmann.ac.il/zemed/davidson_online/mtoh/MTOHeng.html">to play The Magnetic Tower of Hanoi</a>, web applet.
%H A183124 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,3).
%F A183124 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)
%F A183124 a(n)=+4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4), n>=9.
%F A183124 (a(n) = P61(n) as in referenced paper):
%F A183124 a(n) = 3*a(n-1) - 4*n + 18 ; n even ; n >= 5
%F A183124 a(n) = 3*a(n-1) - 4*n + 20  ; n odd ; n >= 6
%F A183124 a(n) = P64(n-1) + P64(n-2) + P75(n-3) + 8*3^(n-4) ; n >= 4
%F A183124 P75(n) and P64(n) refer to the integer sequences described by A122983 and A183120 respectively. See also A183119.
%F A183124 a(n) = (197/324)*3^(n-1) + 2*n - 27/4; n even; n >= 6
%F A183124 a(n) = (197/324)*3^(n-1) + 2*n - 25/4; n odd; n >= 5
%Y A183124 A183122 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.
%Y A183124 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 A183124 Cf. A183111-A183125.
%K A183124 nonn,easy
%O A183124 0,3
%A A183124 _Uri Levy_, Jan 08 2011