A341580 Number of steps needed to reach position "YZ^(n-1)" in the Towers of Hanoi exchanging disks puzzle with 3 pegs and n disks.

0, 1, 3, 6, 12, 23, 44, 82, 153, 284, 528, 979, 1816, 3366, 6241, 11568, 21444, 39747, 73676, 136562, 253129, 469188, 869672, 1611987, 2987920, 5538286, 10265553, 19027816, 35269212, 65373603, 121173924, 224603162, 416315513, 771665884, 1430329248, 2651201459

Offset: 0

Author

Kevin Ryde, Feb 16 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 puzzle is to move a stack of n disks from one peg to another.

Stockmeyer et al. determine the shortest solution to the puzzle (A341579). a(n) is their g(n) which is the number of steps to go from n disks on peg X to the largest on peg Y and the rest on peg Z, denoted "YZ^(n-1)". This is halfway to the solution for n+1 disks since it allows disk n+1 on X to exchange with disk n on Y.

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=2 disks
              *---*         |  A to B
             /     \        |  steps
            *       *       |  a(2) = 3
           / \     / \      |
          *---B---*---*     /
             /     \
        *   /       \   *        n=3 disks
       / \ /         \ / \       A to D
      *---C           *---*      steps
     /     \         /     \     a(3) = 6
    *       *-------*       *
   / \     / \     / \     / \
  *---*---*---D   *---*---*---*
For n=3, the recurrence using A341581 is a(2)=3 from A to B, A341581(2)=2 from D to C, and +1 from B to C.
		

Links

• 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) = g(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. A341579, A341581.

Programs

Mathematica

CoefficientList[Series[x (1+x+x^3)/((1-x)(1-x-x^2-2x^4)),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,-1,2,-2},{0,1,3,6,12},40] (* Harvey P. Dale, Aug 26 2021 *)

PARI

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

Formula

a(n) = a(n-1) + A341581(n-1) + 1, for n>=1. [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^3) /( (1-x) * (1 - x - x^2 - 2*x^4) ).

G.f.: -1/(1-x) + (1 + x + x^2 + x^3)/(1 - x - x^2 - 2*x^4).