A341580 -id:A341580 - cp's OEIS frontend

A341579 Number of steps needed to solve the Towers of Hanoi exchanging disks puzzle with 3 pegs and n disks.

0, 1, 3, 7, 13, 25, 47, 89, 165, 307, 569, 1057, 1959, 3633, 6733, 12483, 23137, 42889, 79495, 147353, 273125, 506259, 938377, 1739345, 3223975, 5975841, 11076573, 20531107, 38055633, 70538425, 130747207, 242347849, 449206325, 832631027, 1543331769, 2860658497

Offset: 0

Kevin Ryde, Feb 15 2021

Scorer, Grundy and Smith define a variation of the towers of Hanoi puzzle where the smallest disk moves freely and two disks can exchange positions when they differ in size by 1, are on different pegs, and each is topmost on its peg. The aim is to move a stack of n disks from one peg to another.

Stockmeyer et al. show that the number of steps in the solution is a(n), and that the sequence of steps is unique. They offer as exercises for the reader to prove that the number of exchanges with the solution is a(n-1), and that a(n) is the largest number of steps between any two configurations (in other words, a(n) is the diameter of the state graph).

			As a graph where each vertex is a configuration of disks on pegs and each edge is a step (as drawn by Scorer et al.),
                A
               / \
              *---*          n=3 disks
             /     \          A to D
            *       *      a(3) = 7 steps
           / \     / \
          *---B---*---*
             /     \
        *   /       \   *
       / \ /         \ / \
      *---C           *---*
     /     \         /     \
    *       *-------*       *
   / \     / \     / \     / \
  D---*---*---*   *---*---*---*
The formula using A341580 is A to B distance A341580(2) = 3, the same (by symmetry) from D to C, and +1 from B to C.  B to C is where the largest disk moves to the target peg (by exchange with the second-largest).

Kevin Ryde, Table of n, a(n) for n = 0..700
House of Graphs, graphs 44105, 44107, 44109. Diameters = a(3..5).
R. S. Scorer, P. M. Grundy, and C. A. B. Smith, Some Binary Games, The Mathematical Gazette, July 1944, volume 28, number 280, pages 96-103, section 4(iii) Plane Network Game.
Paul K. Stockmeyer, C. Douglass Bateman, James W. Clark, Cyrus R. Eyster, Matthew T. Harrison, Nicholas A. Loehr, Patrick J. Rodriguez, and Joseph R. Simmons III, Exchanging Disks in the Tower of Hanoi, International Journal of Computer Mathematics, volume 59, number 1-2, pages 37-47, 1995. Also author's copy. a(n) = f(n) in section 3.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2,-2).
Index entries for sequences related to Towers of Hanoi

Cf. A341580 (halfway), A341582 (first differences), A341583 (geometric length).

A341579list[nmax_]:=LinearRecurrence[{2,0,-1,2,-2},{0,1,3,7,13},nmax+1];A341579list[50] (* Paolo Xausa, Jun 29 2023 *)

my(p=Mod('x,'x^4-'x^3-'x^2-2)); a(n) = subst(lift(p^n),'x,2) - 1;

a(n) = 2*A341580(n-1) + 1 for n>=1. [Stockmeyer et al.]
a(n) = a(n-1) + a(n-2) + 2*a(n-4) + 3 for n>=4. [Stockmeyer et al.]
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) - 2*a(n-5).
G.f.: x*(1 + x + x^2) / ( (1-x) * (1 - x - x^2 - 2*x^4) ).
G.f.: -1/(1-x) + (1 + x + x^2 + 2*x^3)/(1 - x - x^2 - 2*x^4).

A341581 Number of steps needed to move the largest disk out from a stack of n disks in the Towers of Hanoi exchanging disks puzzle with 3 pegs.

Original entry on oeis.org

0, 1, 2, 5, 10, 20, 37, 70, 130, 243, 450, 836, 1549, 2874, 5326, 9875, 18302, 33928, 62885, 116566, 216058, 400483, 742314, 1375932, 2550365, 4727266, 8762262, 16241395, 30104390, 55800320, 103429237, 191712350, 355350370, 658663363, 1220872210, 2262960276

Offset: 0

Kevin Ryde, Feb 16 2021

Scorer, Grundy and Smith define a variation of the towers of Hanoi puzzle where the smallest disk moves freely and two disks can exchange positions when they differ in size by 1, are on different pegs, and each is topmost on its peg. The puzzle is to move a stack of n disks from one peg to another.

Stockmeyer et al. determine the shortest solution to the puzzle. a(n) is their h(n) which is the number of steps to go from n disks on peg X to the largest disk to peg Y and the others remaining on X. This arises in A341580 to go between subgraph "connection" points.

			As a graph where each vertex is a configuration of disks on pegs and each edge is a step (as drawn by Scorer et al.),
                A
               / \
              *---*         n=3 disks
             /     \         A to D
            *       *        steps
           / \     / \      a(3) = 5
          *---B---*---*
             /     \
        D   /       \   *
       / \ /         \ / \
      *---C           *---*
     /     \         /     \
    *       *-------*       *
   / \     / \     / \     / \
  *---*---*---*   *---*---*---*
The recurrence using A341579 and A341580 is steps A341580(2)=3 from A to B, +1 from B to C, and A341579(1) = 1 from C to D (the whole puzzle solution in an n-2 subgraph).

Kevin Ryde, Table of n, a(n) for n = 0..700
Paul K. Stockmeyer et al., Exchanging Disks in the Tower of Hanoi, International Journal of Computer Mathematics, volume 59, number 1-2, pages 37-47, 1995. Also author's copy. a(n) = h(n) in section 3.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2,-2).
Index entries for sequences related to Towers of Hanoi

Cf. A341579, A341580.

my(p=Mod('x,'x^4-'x^3-'x^2-2)); a(n) = subst(lift(p^(n+2))\'x,'x,2)/2 - 1;

a(n) = A341579(n-2) + A341580(n-1) + 1, for n>=2. [Stockmeyer et al.]
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) - 2*a(n-5).
G.f.: x * (1 + x^2 + x^3) /( (1-x) * (1 - x - x^2 - 2*x^4) ).
G.f.: -1/(1-x) + (1 + x + x^3)/(1 - x - x^2 - 2*x^4).

A341582 Number of simple moves of the smallest disk in the solution to the Towers of Hanoi exchanging disks puzzle with 3 pegs and n disks.

Original entry on oeis.org

0, 1, 2, 4, 6, 12, 22, 42, 76, 142, 262, 488, 902, 1674, 3100, 5750, 10654, 19752, 36606, 67858, 125772, 233134, 432118, 800968, 1484630, 2751866, 5100732, 9454534, 17524526, 32482792, 60208782, 111600642, 206858476, 383424702, 710700742, 1317326728, 2441744422

Offset: 0

Kevin Ryde, Feb 16 2021

Scorer, Grundy and Smith define a variation of the Towers of Hanoi puzzle where the smallest disk moves freely and two disks can exchange positions when they differ in size by 1, are on different pegs, and each is topmost on its peg. The puzzle is to move a stack of n disks from one peg to another.

Stockmeyer et al. show the shortest solution to the puzzle is f(n) = A341579(n) steps. They offer as an exercise for the reader to prove that the number of exchanges made within the solution is f(n-1) and the remaining a(n) = f(n) - f(n-1) here is the number of simple moves of the smallest disk (move-only, not exchange).

			As a graph where each vertex is a configuration of disks on pegs and each edge is a step (as drawn by Scorer et al.),
                A
               / \              n=3 disks
              B---*             solution
             /     \            A to H,
            C       *           within
           / \     / \          which
          *---D---*---*         a(3) = 4
             /     \            smallest
        *   /       \   *       disk moves:
       / \ /         \ / \       AB, CD,
      F---E           *---*      EF, GH
     /     \         /     \
    G       I-------*       *
   / \     / \     / \     / \
  H---*---*---J   *---*---*---*
For n=4, the first half of the solution is A to J per A341580.  The smallest disk moves are AB, CD, IJ, and twice those is a(4) = 2*3 = 6 since J across to the next subgraph is an exchange, not a smallest disk move.

Kevin Ryde, Table of n, a(n) for n = 0..700
Paul K. Stockmeyer et al., Exchanging Disks in the Tower of Hanoi, International Journal of Computer Mathematics, volume 59, number 1-2, pages 37-47, 1995. Also author's copy. See section 5 exercise 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,2).
Index entries for sequences related to Towers of Hanoi

Cf. A341579, A341580.

A341582list[nmax_]:=LinearRecurrence[{1,1,0,2},{0,1,2,4},nmax+1];A341582list[50] (* Paolo Xausa, Jun 29 2023 *)

my(p=Mod('x,'x^4-'x^3-'x^2-2)); a(n) = subst(lift(p^n)\'x,'x,2);

a(n) = A341579(n) - A341579(n-1), for n>=1. [Stockmeyer et al.]
a(n) = a(n-1) + a(n-2) + 2*a(n-4).
G.f.: x*(1 + x + x^2)/(1 - x - x^2 - 2*x^4)

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A341579 Number of steps needed to solve the Towers of Hanoi exchanging disks puzzle with 3 pegs and n disks.

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A341581 Number of steps needed to move the largest disk out from a stack of n disks in the Towers of Hanoi exchanging disks puzzle with 3 pegs.

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A341582 Number of simple moves of the smallest disk in the solution to the Towers of Hanoi exchanging disks puzzle with 3 pegs and n disks.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula