A183119 -id:A183119 - cp's OEIS frontend

A183125 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

0, 1, 4, 11, 30, 83, 236, 687, 2026, 6027, 18008, 53927, 161654, 484803, 1454212, 4362399, 13086914, 39260411, 117780848, 353342103, 1060025806, 3180076851, 9540229916, 28620689039, 85862066330, 257586198123, 772758593416, 2318275779207, 6954827336486, 20864482008227, 62593446023348, 187780338068607, 563341014204274, 1690023042611163

Offset: 0

Uri Levy, Jan 08 2011

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. 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 link 2 listed below.

Large N limit of the sequence is 0.5*(197/324)*3^N ~ 0.5*0.61*3^N. Series designation: S61(n).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Muniru A Asiru, Table of n, a(n) for n = 0..2020
Uri Levy, The Magnetic Tower of Hanoi, arxiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet [Broken link]
Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,7,-3).

A183123 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.
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.
Cf. A183111 - A183125.

a:=[30, 83, 236, 687, 2026];; for n in [6..30] do a[n]:=5*a[n-1]-6*a[n-2] -2*a[n-3]+7*a[n-4]-3*a[n-5]; od; Concatenation([0, 1, 4, 11], a); # G. C. Greubel, Dec 04 2018

I:=[0,1,4,11,30,83,236,687,2026]; [n le 9 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

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( (-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1))); // G. C. Greubel, Dec 04 2018

seq(coeff(series((-4*x^8-2*x^6+x^4-3*x^3-x^2+x)/(3*x^5-7*x^4+2*x^3+6*x^2-5*x+1),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Dec 04 2018

Join[{0, 1, 4, 11}, LinearRecurrence[{5, -6, -2, 7, -3}, {30, 83, 236, 687, 2026}, 30]] (* Jean-François Alcover, Dec 04 2018 *)
CoefficientList[Series[(- 4 x^8 - 2 x^6 + x^4 - 3 x^3 - x^2 + x) / (3 x^5 - 7 x^4 + 2 x^3 + 6 x^2 - 5 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Dec 04 2018 *)

my(x='x+O('x^30)); concat([0], Vec((-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1))) \\ G. C. Greubel, Dec 04 2018

s=((-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1)).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 04 2018

G.f.: (-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1).
a(n) = +5*a(n-1)-6*a(n-2)-2*a(n-3)+7*a(n-4)-3*a(n-5).
(a(n) = S61(n) as in referenced paper):
a(n) = 3*a(n-1) - 2*n^2 + 17*n - 43 ; n even ; n >= 6.
a(n) = 3*a(n-1) - 2*n^2 + 17*n - 42 ; n odd ; n >= 5.
a(n) = S64(n-1) + S64(n-2) + S75(n-3) + 4*3^(n-3) + 2 ; n >= 3.
S64(n) and S75(n) refer to the integer sequences described by A183121 and A183119 respectively.
a(n) = 0.5*(197/324)*3^n + n^2 - 5.5*n + 91/8; n even; n >= 4.
a(n) = 0.5*(197/324)*3^n + n^2 - 5.5*n + 93/8; n odd; n >= 5.

More terms from Jean-François Alcover, Dec 04 2018

A122983 a(n) = (2 + (-1)^n + 3^n)/4.

Original entry on oeis.org

1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581, 1195743, 3587227, 10761681, 32285041, 96855123, 290565367, 871696101, 2615088301, 7845264903, 23535794707, 70607384121, 211822152361, 635466457083

Offset: 0

Paul Barry, Sep 22 2006

Old definition was: "Binomial transform of aeration of A081294".
Binomial transform is A063376.
A122983 = (1,1,3,7,1,1,3,7,...) mod 10. - M. F. Hasler, Feb 25 2008
Equals row sums of triangle A158301. - Gary W. Adamson, Mar 15 2009
a(n) = the number of ternary sequences of length n where the numbers of (0's, 1's) are both even. A015518 covers the (odd, even) and (even, odd) cases, and A081251 covers (odd, odd). - Toby Gottfried, Apr 18 2010
This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi (MToH) puzzle. The sequence A183119 is the partial sums of the sequence in question (obviously describing the total number of moves associated with the specific solution algorithm). For other MToH-related sequences, Cf. A183111 - A183125.
Let B=[1,sqrt(2),0; sqrt(2),1,sqrt(2); 0,sqrt(2),1] be a 3 X 3 matrix. Then a(n)=[B^n](1,1), n=0,1,2,.... - _L. Edson Jeffery, Dec 21 2011
Also the domination number of the n-Hanoi graph. - Eric W. Weisstein, Jun 16 2017
Also the matching number of the n-Sierpinski gasket graph. - Eric W. Weisstein, Jun 17 2017
Let M = [1,1,1,0; 1,1,0,1; 1,0,1,1; 0,1,1,1], a 4 X 4 matrix. Then a(n) is the upper left entry in M^n. - Philippe Deléham, Aug 23 2020
Also the lower matching number (=independent domination number) of the n-Hanoi graph. - Eric W. Weisstein, Aug 01 2023

M. F. Hasler, Table of n, a(n) for n = 0..199.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin, Peaks Sets of Classical Coxeter Groups, arXiv preprint, arXiv:1505.04479 [math.GR], 2015.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 99. Book's website
Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Hanoi Graph.
Eric Weisstein's World of Mathematics, Lower Independence Number.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. a(j+1) = A137822(2^j) and these are the record values of A137822.

Cf. A054879 (bisection), A066443 (bisection). Row sums of A158303.

A122983 := n -> ceil(3^n/4); 'A122983(n)' $ n=0..22; # M. F. Hasler, Feb 25 2008
a[ -1]:=1:a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-2 od: seq(a[n], n=-1..25); # Zerinvary Lajos, Apr 28 2008

CoefficientList[Series[(1 - 2 x - x^2)/((1 - x) (1 + x) (1 - 3 x)), {x, 0, 40}], x] (* Harvey P. Dale, Sep 03 2013 *)
LinearRecurrence[{3, 1, -3}, {1, 1, 3}, 40] (* Harvey P. Dale, Sep 03 2013 *)
Table[(2 + (-1)^n + 3^n)/4, {n, 0, 20}] (* Eric W. Weisstein, Jun 16 2017 *)
Table[Floor[3^n/4] + 1, {n, 0, 20}] (* Eric W. Weisstein, Jan 17 2018 *)
Floor[3^Range[0, 20]/4] + 1 (* Eric W. Weisstein, Jan 17 2018 *)

A122983(n)=3^n\4+1 \\ M. F. Hasler, Feb 25 2008

def A122983(n): return (1 if n&1 else 3)+3**n>>2 # Chai Wah Wu, Apr 12 2023

From Paul Barry, Jun 14 2007: (Start)
G.f.: (1-2*x-x^2)/((1-x)*(1+x)*(1-3*x));
a(n) = 3^n/4+(-1)^n/4+1/2;
E.g.f.: cosh(x)^2*exp(x). (End)
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Sep 03 2013
E.g.f.: Q(0)/2, where Q(k) = 1 + 3^k/( 2 - 2*(-1)^k/( 3^k + (-1)^k - 2*x*3^k/( 2*x + (k+1)*(-1)^k/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2013
a(2*n) = 3*a(2*n-1); a(2*n+1) = 3*a(2*n) - 2. - Philippe Deléham, Aug 23 2020

Extended and corrected (existing Maple code) by M. F. Hasler, Feb 25 2008

Description changed to formula by Eric W. Weisstein, Jun 16 2017

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.

Original entry on oeis.org

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

Offset: 0

Uri Levy, Jan 05 2011

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.
Disk numbering is from largest disk (k = 1) to smallest disk (k = N)
The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
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).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

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.
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.
Cf. A183111-A183125.

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 *)

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(n) = 4*a(n-1)- 2*a(n-2)-4*a(n-3)+3*a(n-4). [Harvey P. Dale, May 04 2012]

More terms from Harvey P. Dale, May 04 2012

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.

Original entry on oeis.org

0, 1, 4, 11, 30, 85, 244, 715, 2118, 6317, 18900, 56635, 169822, 509365, 1527972, 4583771, 13751142, 41253229, 123759460, 371278123, 1113834078, 3341501909, 10024505364, 30073515691, 90220546630, 270661639405, 811984917684, 2435954752475, 7307864256798, 21923592769717, 65770778308420, 197312334924475

Offset: 0

Uri Levy, Jan 05 2011

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.

Large N limit of the sequence is 0.5*(23/36)*3^N =~ 0.5*0.64*3^N. Series designation: S64(n).

U. Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843, 2010.
U. Levy, to play The Magnetic Tower of Hanoi, web applet. [Broken link]
Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,7,-3).

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.
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.
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.
Cf. A183111 - A183125.

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

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

Join[{0, 1}, LinearRecurrence[{5, -6, -2, 7, -3}, {4, 11, 30, 85, 244}, 30]] (* Jean-François Alcover, Dec 04 2018 *)
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 *)

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

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

G.f.: x*(1-x-3*x^2+x^3+2*x^4-4*x^5)/((1+x)*(1-3*x)*(1-x)^3).
(a(n) = S64(n) as in referenced paper):
a(n) = 3*a(n-1) - n^2 + 6*n - 11; n even; n >= 4.
a(n) = 3*a(n-1) - n^2 + 6*n - 10; n odd; n >= 3.
a(n) = a(n-1) + 2* S75(n-3) + 5*3^(n-3) + 2; n >= 3
S75(n) refers to the integer sequence described by A183119.
a(n) = 0.5*(23/36)*3^n + 0.5*n^2 - 1.5*n + 17/8; n even; n >= 2.
a(n) = 0.5*(23/36)*3^n + 0.5*n^2 - 1.5*n + 19/8; n odd; n >= 3.
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

More terms from Jean-François Alcover, Dec 04 2018

A183122 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 [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

Original entry on oeis.org

0, 1, 3, 7, 19, 53, 153, 455, 1359, 4073, 12213, 36635, 109899, 329693, 989073, 2967215, 8901639, 26704913, 80114733, 240344195, 721032579, 2163097733, 6489293193, 19467879575, 58403638719, 175210916153, 525632748453, 1576898245355, 4730694736059

Offset: 0

Uri Levy, Jan 07 2011

nonn

A.   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. The solution algorithm producing the presented sequence is NOT optimal. The particular "62" algorithm solving the puzzle at hand is presented and discussed in the paper referenced by link 1 below. 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 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 (67/108)*3^(k-1) ~ 0.62*3^(k-1). Series designation: P62(k).

U. Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225
U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arxiv:1011.3843
U. Levy, to play The Magnetic Tower of Hanoi, web applet
Index entries for linear recurrences with constant coefficients, signature (3, 1, -3).

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 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

Join[{0,1,3,7},LinearRecurrence[{3,1,-3},{19,53,153},30]] (* Harvey P. Dale, Dec 08 2014 *)

a(n)=+3*a(n-1)+a(n-2)-3*a(n-3) for n>6.
g.f.: x+ 3*x^2 +7*x^3 -x^4*(-19+4*x+25*x^2)/ ((x-1)(3*x-1)(1+x)).
(a(n) = P62(n) as in referenced paper):
a(n) = 3*a(n-1) - 6; n even; n >= 6
a(n) = 3*a(n-1) - 4; n odd; n >= 5
a(n) = P67(n-1) + P67(n-2) + P75(n-3) + 8*3^(n-4) ; n >= 4
P75(n) and P67(n) refer to the integer sequences described by A122983 and A100702 respectively. See also A183119.
a(n) = (67/108)*3^(n-1) + 9/4; n even; n >= 4
a(n) = (67/108)*3^(n-1) + 11/4; n odd; n >= 5

More terms from Harvey P. Dale, Dec 08 2014

A183123 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

Original entry on oeis.org

0, 1, 4, 11, 30, 83, 236, 691, 2050, 6123, 18336, 54971, 164870, 494563, 1483636, 4450851, 13352490, 40057403, 120172136, 360516331, 1081548910, 3244646643, 9733939836, 29201819411, 87605458130, 262816374283, 788449122736, 2365347368091, 7096042104150

Offset: 0

Uri Levy, Jan 07 2011

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. The solution algorithm producing the presented sequence is NOT optimal. The particular "62" algorithm solving the puzzle at hand is presented and discussed in the paper referenced by link 1 below. 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 link 2 listed below.

Large N limit of the sequence is 0.5*(67/108)*3^N ~ 0.5*0.62*3^N. Series designation: S62(n).

U. Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
U. Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arxiv:1011.3843 [math.CO], 2010.
U. Levy, to play The Magnetic Tower of Hanoi, web applet.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Cf. A183122 - "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 [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle" is an "original" sequence describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "62" algorithm mentioned above.

Cf. 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. A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

LinearRecurrence[{4,-2,-4,3},{0,1,4,11,30,83,236},40] (* Harvey P. Dale, Jun 07 2015 *)

concat(0, Vec(x*(4*x^5+2*x^4+2*x^3+3*x^2-1)/((x-1)^2*(x+1)*(3*x-1)) + O(x^100))) \\ Colin Barker, Sep 18 2014

a(n)=+4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4) for n>6.
(a(n) = S62(n) as in referenced paper):
S62(n) = 3*S62(n-1) - 5*n + 17; n even; n >= 4
S62(n) = 3*S62(n-1) - 5*n + 13; n odd; n >= 5
S62(n) = S67(n-1) + S67(n-2) + S75(n-3) + 4*3^(n-3) + 2; n >= 3
S67(n) and S75(n) refer to the integer sequences described by A104743 and A183119 respectively.
S62(n) = 0.5*(67/108)*3^n + 2.5*n - 41/8; n even; n >= 4
S62(n) = 0.5*(67/108)*3^n + 2.5*n - 39/8; n odd; n >= 3.
a(n) = -5-(-1)^n/8+(67*3^(-3+n))/8+(5*n)/2 for n>2. - Colin Barker, Sep 18 2014
G.f.: x*(4*x^5+2*x^4+2*x^3+3*x^2-1) / ((x-1)^2*(x+1)*(3*x-1)). - Colin Barker, Sep 18 2014

More terms and correction to recurrence by Colin Barker, Sep 18 2014

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.

Original entry on oeis.org

0, 1, 3, 7, 19, 53, 153, 451, 1339, 4001, 11981, 35919, 107727, 323149, 969409, 2908187, 8724515, 26173497, 78520437, 235561255, 706683703

Offset: 0

Uri Levy, Jan 08 2011

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.
Disk numbering is from largest disk (k = 1) to smallest disk (k = N).
The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
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).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

A183122 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.
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.
Cf. A183111-A183125.

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)
a(n)=+4*a(n-1)-2*a(n-2)-4*a(n-3)+3*a(n-4), n>=9.
(a(n) = P61(n) as in referenced paper):
a(n) = 3*a(n-1) - 4*n + 18 ; n even ; n >= 5
a(n) = 3*a(n-1) - 4*n + 20  ; n odd ; n >= 6
a(n) = P64(n-1) + P64(n-2) + P75(n-3) + 8*3^(n-4) ; n >= 4
P75(n) and P64(n) refer to the integer sequences described by A122983 and A183120 respectively. See also A183119.
a(n) = (197/324)*3^(n-1) + 2*n - 27/4; n even; n >= 6
a(n) = (197/324)*3^(n-1) + 2*n - 25/4; n odd; n >= 5

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A183125 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

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A122983 a(n) = (2 + (-1)^n + 3^n)/4.

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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.

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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.

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A183122 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 [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

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A183123 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

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