OFFSET
0,3
COMMENTS
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).
REFERENCES
Uri 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..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).
FORMULA
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.
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.
MAPLE
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
MATHEMATICA
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 *)
PROG
(Magma) 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
(PARI) 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
(Magma) m:=30; R<x>:=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
(Sage) 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
(GAP) 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
CROSSREFS
A183123 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.
KEYWORD
nonn,easy
AUTHOR
Uri Levy, Jan 08 2011
EXTENSIONS
More terms from Jean-François Alcover, Dec 04 2018
STATUS
approved