cp's OEIS Frontend

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.

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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.

Original entry on oeis.org

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

Views

Author

Uri Levy, Jan 08 2011

Keywords

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

Crossrefs

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.

Programs

  • 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
  • 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
  • Magma
    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
  • 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 *)
  • 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
  • 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

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.
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.

Extensions

More terms from Jean-François Alcover, Dec 04 2018

A104743 a(n) = 3^n + n.

Original entry on oeis.org

1, 4, 11, 30, 85, 248, 735, 2194, 6569, 19692, 59059, 177158, 531453, 1594336, 4782983, 14348922, 43046737, 129140180, 387420507, 1162261486, 3486784421, 10460353224, 31381059631, 94143178850, 282429536505, 847288609468
Offset: 0

Views

Author

Zak Seidov, Mar 23 2005

Keywords

Comments

This sequence also describes the total number of moves solving (non-optimally) the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (see the "CROSSREFS" in A183121). For other Magnetic Tower of Hanoi related sequences, cf. A183111 - A183125.
Form an infinite array with m(0,k) = 2*k+1 and m(n,k) = Sum_{r=0..n-1} m(r,k) + Sum_{c=0..k-1} m(n,c), being the sum of the terms above m(n,k) plus those terms to the left of m(n,k). The first row is A005408. The second row is A033484. The first column is A011782. The sum of the terms in the n-th antidiagonal is a(n). - J. M. Bergot, Jun 07 2013

Links

Crossrefs

Cf. A103537.

Programs

Formula

a(n) = n + 3^n.
From Colin Barker, Jan 25 2012: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
G.f.: (1+x)*(1-2*x) / ((1-3*x)*(1-x)^2). (End)
E.g.f.: x*exp(x) + exp(3*x). - G. C. Greubel, May 21 2019

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Jan 18 2009
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