cp's OEIS Frontend

A183121 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.

%I A183121 #33 Sep 08 2022 08:45:55
%S A183121 0,1,4,11,30,85,244,715,2118,6317,18900,56635,169822,509365,1527972,
%T A183121 4583771,13751142,41253229,123759460,371278123,1113834078,3341501909,
%U A183121 10024505364,30073515691,90220546630,270661639405,811984917684,2435954752475,7307864256798,21923592769717,65770778308420,197312334924475
%N A183121 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.
%C A183121 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.
%C A183121 Large N limit of the sequence is 0.5*(23/36)*3^N =~ 0.5*0.64*3^N. Series designation: S64(n).
%D A183121 U. Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
%H A183121 Muniru A Asiru, <a href="/A183121/b183121.txt">Table of n, a(n) for n = 0..2000</a>
%H A183121 U. Levy, <a href="http://arxiv.org/abs/1003.0225">The Magnetic Tower of Hanoi</a>, arXiv:1003.0225 [math.CO], 2010.
%H A183121 U. Levy, <a href="http://arxiv.org/abs/1011.3843">Magnetic Towers of Hanoi and their Optimal Solutions</a>, arXiv:1011.3843, 2010.
%H A183121 U. Levy, <a href="http://www.weizmann.ac.il/zemed/davidson_online/mtoh/MTOHeng.html">to play The Magnetic Tower of Hanoi</a>, web applet. [Broken link]
%H A183121 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-2,7,-3).
%F A183121 G.f.: x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/((1+x)*(1-3*x)*(1-x)^3).
%F A183121 (a(n) = S64(n) as in referenced paper):
%F A183121 a(n) = 3*a(n-1) - n^2 + 6*n - 11; n even; n >= 4.
%F A183121 a(n) = 3*a(n-1) - n^2 + 6*n - 10; n odd; n >= 3.
%F A183121 a(n) = a(n-1) + 2* S75(n-3) + 5*3^(n-3) + 2; n >= 3
%F A183121 S75(n) refers to the integer sequence described by A183119.
%F A183121 a(n) = 0.5*(23/36)*3^n + 0.5*n^2 - 1.5*n + 17/8; n even; n >= 2.
%F A183121 a(n) = 0.5*(23/36)*3^n + 0.5*n^2 - 1.5*n + 19/8; n odd; n >= 3.
%F A183121 a(n) = 5*a(n-1)-6*a(n-2)-2*a(n-3)+7*a(n-4)-3*a(n-5), for n>5. - _Vincenzo Librandi_, Dec 04 2018
%p A183121 seq(coeff(series(x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/((1+x)*(1-3*x)*(1-x)^3)), x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Dec 04 2018
%t A183121 Join[{0, 1}, LinearRecurrence[{5, -6, -2, 7, -3}, {4, 11, 30, 85, 244}, 30]] (* _Jean-François Alcover_, Dec 04 2018 *)
%t A183121 CoefficientList[Series[x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/((1+x)*(1-3*x)*(1-x)^3), {x, 0, 33}], x] (* _Vincenzo Librandi_, Dec 04 2018 *)
%o A183121 (Magma) I:=[0,1,4,11,30,85,244]; [n le 7 select I[n] else 5*Self(n-1)-6*Self(n-2)-2*Self(n-3)+7*Self(n-4)-3*Self(n-5): n in [1..35]]; // _Vincenzo Librandi_, Dec 04 2018
%o A183121 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/ ((1+x)*(1-3*x)*(1-x)^3))) \\ _G. C. Greubel_, Dec 04 2018
%o A183121 (Sage) s=(x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/((1+x)*(1-3*x)*(1-x)^3) ).series(x, 30); s.coefficients(x, sparse=False) # _G. C. Greubel_, Dec 04 2018
%Y A183121 A183120 - is an "original" sequence describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "64" algorithm mentioned above.
%Y A183121 A104743 - is a sequence also describing the total number of moves, 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(n) = a(n-1) + 2*3^(n-2) + 1 and the closed-form expression is 3^(n-1) + n - 1. Large N limit is 0.5*(2/3)*3^N =~ 0.5*0.67*3^N, and sequence designation is thus S67(n). The (non-optimal) "67" algorithm solving the Magnetic Tower of Hanoi with the given pre-coloring configuration yielding the S67(n) sequence (given by A104743) is explicitly described and discussed in the paper referenced in link 1 above.
%Y A183121 A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.
%Y A183121 Cf. A183111 - A183125.
%K A183121 nonn,easy
%O A183121 0,3
%A A183121 _Uri Levy_, Jan 05 2011
%E A183121 More terms from _Jean-François Alcover_, Dec 04 2018