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

Author

Kevin Ryde, Feb 15 2021

Keywords

• nonn
• easy

Comments

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).

Examples

			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).
		

Links

• 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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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).