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.

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

Author

Uri Levy, Jan 05 2011

Keywords

• nonn
• easy

Comments

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).

References

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

Links

• 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).

Crossrefs

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.

Programs

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

Maple

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

Mathematica

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 *)

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

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

Formula

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

Extensions

More terms from Jean-François Alcover, Dec 04 2018