This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A183120 #27 May 06 2024 06:54:05
%S A183120 0,1,3,7,19,55,159,471,1403,4199,12583,37735,113187,339543,1018607,
%T A183120 3055799,9167371,27502087,82506231,247518663,742555955,2227667831,
%U A183120 6683003455,20049010327,60147030939,180441092775,541323278279,1623969834791,4871909504323
%N 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.
%C A183120 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 A183120 Disk numbering is from largest disk (k = 1) to smallest disk (k = N)
%C A183120 The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
%C A183120 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).
%D A183120 Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.
%H A183120 Uri Levy, <a href="http://arxiv.org/abs/1003.0225">The Magnetic Tower of Hanoi</a>, arXiv:1003.0225 [math.CO], 2010.
%H A183120 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 A183120 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 A183120 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,3).
%F A183120 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.
%F A183120 (a(n) = P64(n) as in referenced paper):
%F A183120 a(n) = 3*a(n-1) - 2*n + 6; n even; n >= 4
%F A183120 a(n) = 3*a(n-1) - 2*n + 8; n odd; n >= 5
%F A183120 a(n) = a(n-1) + 2* P75(n-3) + 10*3^(n-4); n >= 4
%F A183120 P75(n) refers to the integer sequence described by A122983. See also A183119.
%F A183120 a(n) = (23/36)*3^(n-1) + n - 9/4; n even; n >= 4
%F A183120 a(n) = (23/36)*3^(n-1) + n - 7/4; n odd; n >= 3
%F A183120 a(n) = 4*a(n-1)- 2*a(n-2)-4*a(n-3)+3*a(n-4). [_Harvey P. Dale_, May 04 2012]
%t A183120 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 *)
%Y A183120 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.
%Y A183120 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.
%Y A183120 Cf. A183111-A183125.
%K A183120 nonn,easy
%O A183120 0,3
%A A183120 _Uri Levy_, Jan 05 2011
%E A183120 More terms from _Harvey P. Dale_, May 04 2012