A007664 Reve's puzzle: number of moves needed to solve the Towers of Hanoi puzzle with 4 pegs and n disks, according to the Frame-Stewart algorithm.

0, 1, 3, 5, 9, 13, 17, 25, 33, 41, 49, 65, 81, 97, 113, 129, 161, 193, 225, 257, 289, 321, 385, 449, 513, 577, 641, 705, 769, 897, 1025, 1153, 1281, 1409, 1537, 1665, 1793, 2049, 2305, 2561, 2817, 3073, 3329, 3585, 3841, 4097, 4609, 5121, 5633

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Keywords

• nonn
• nice

Comments

The Frame-Stewart algorithm minimizes the number of moves a(n) needed to first move k disks to an intermediate peg (requiring a(k) moves), then moving the remaining n-k disks to the destination peg without touching the k smallest disks (requiring 2^(n-k)-1 moves) and finally moving the k smaller disks to the destination.

This leads to the given recursive formula a(n) = min{...}. It follows that the sequence of first differences is A137688 = (1,2,2,4,4,4,...) = 2^A003056(n), which in turn gives the explicit formulas for a(n) as partial sums of A137688.

"Numerous others have rediscovered this algorithm over the years [several references omitted]; many of these failed to derive the correct value for the parameter i, most mistakenly thought that they had actually proved optimality and almost none contributed anything new to what was done by Frame and Stewart". [Stockmeyer]

Numbers of the form 2^k+1 appear for n = 2, 3, 4, 6, 8, 11, 15, 15+4 = 19, 19+5 = 24, 24+6 = 30, 30+7 = 37, 37+8 = 45, ... - Max Alekseyev, Feb 06 2008

The Frame-Stewart algorithm indeed gives the optimal solution, i.e., the minimal possible number of moves for the case of four pegs [Bousch, 2014]. - Andrey Zabolotskiy, Sep 18 2017

References

• A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math., 8 (1975-1976), 169-176.
• Paul Cull and E. F. Ecklund, On the Towers of Hanoi and generalized Towers of Hanoi problems. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 229-238. MR0725883(85a:68059).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• D. Wood, Towers of Brahma and Hanoi revisited, J. Recreational Math., 14 (1981), 17-24.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..10012 (first 1001 terms from M. F. Hasler)
• S. Alejandre, Legend of Towers of Hanoi.
• J.-P. Allouche, Note on the cyclic towers of Hanoi, Theoret. Comput. Sci., 123 (1994), 3-7.
• T. Bousch, La quatrième tour de Hanoi, Bull. Belg. Math. Soc. Simon Stevin 21 (2014) 895-912.
• A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math 8.3 (1975-6), 169-176. (Annotated scanned copy)
• A. M. Hinz, An iterative algorithm for the Tower of Hanoi with four pegs, Computing, June 1989, Volume 42, Issue 2-3, pp. 133-140.
• A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013.
• B. Houston and H. Masum, Explorations in 4-peg Tower of Hanoi. [Paper]
• B. Houston and H. Masum, Explorations in 4-peg Tower of Hanoi. [Web site]
• S. Klavzar et al., Hanoi graphs and some classical numbers.
• S. Klavzar and U. Milutinovic, Simple explicit formulas for the Frame-Stewart's numbers.
• S. Klavzar, U. Milutinovic and C. Petr, On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem, Discrete Appl. Math. 120, 1-3 (2002), 141 - 157.
• Mathnet at U. Toronto, Generalizing the Towers of Hanoi Problem.
• Richard E. Korf and Ariel Felner, Recent Progress in Heuristic Search: a Case Study of the Four-Peg Towers of Hanoi Problem. IJCAI 2007: 2324-2329.
• B. M. Stewart, Advanced Problem 3918, Amer. Math. Monthly, 46 (1939), 363.
• B. M. Stewart & J. S. Frame, Solution to Problem 3918, Amer. Math. Monthly, 48 (1941), 217-219.
• P. Stockmeyer, Variations on the Four-Post Tower of Hanoi Puzzle, Congressus Numerantium 102 (1994), pp. 3-12. [Has extensive bibliography]
• Eric Weisstein's World of Mathematics, Towers of Hanoi.
• Janez Žerovnik, Self Similarities of the Tower of Hanoi Graphs and a proof of the Frame-Stewart Conjecture, arXiv:1601.04298 [math.CO], 2016.
• Index entries for sequences related to Towers of Hanoi

Crossrefs

Cf. A007665, A182058, A003056, A000225 (analog for 3 pegs), A137688 (first differences).

Programs

Haskell

a007664 = sum . map (a000079 . a003056) . enumFromTo 0 . subtract 1
-- Reinhard Zumkeller, Feb 17 2013

Maple

A007664:=proc(n) option remember;min(seq(2*A007664(k)+2^(n-k)-1,k=0..n-1)) end; A007664(0):=0; # M. F. Hasler, Feb 06 2008
A007664 := n -> 1 + (n - 1 - A003056(n)*(A003056(n) - 1)/2)*2^A003056(n); A003056 := n -> round(sqrt(2*n+2))-1; # M. F. Hasler, Feb 06 2008

Mathematica

a[n_] := a[n] = Min[ Table[ 2*a[k] + 2^(n-k) - 1, {k, 0, n-1}]]; a[0] = 0; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Dec 06 2011, after M. F. Hasler *)
Join[{0},Accumulate[2^Flatten[Table[PadRight[{},n+1,n],{n,0,12}]]]] (* Harvey P. Dale, Jul 03 2021 *)

PARI

A007664(n) = (n - 1 - (n=A003056(n))*(n-1)/2)*2^n +1
A003056(n) = (sqrt(2*n+2)-.5)\1 \\ M. F. Hasler, Feb 06 2008

PARI

print_7664(n,s=0,t=1,c=1,d=1)=while(n-->=0,print1(s+=t,",");c--&next;c=d++;t<<=1)

PARI

A007664(n,c=1,d=1,t=1)=sum(i=c,n,i>c&(t<<=1)&c+=d++;t) \\ M. F. Hasler, Feb 06 2008

Python

from math import isqrt
def A007664(n): return (1<<(r:=(k:=isqrt(m:=n+1<<1))+int(m>=k*(k+1)+1)-1))*(n-1-(r*(r-1)>>1))+1 # Chai Wah Wu, Oct 17 2022

Formula

a(n) = min{ 2 a(k) + 2^(n-k) - 1; k < n}, which is always odd. - M. F. Hasler, Feb 06 2008

a(n) = Sum_{i=0..n-1} 2^A003056(i). - Daniele Parisse, May 09 2003

a(n) = 1 + (n + A003056(n) - 1 - A003056(n)*(A003056(n) + 1)/2)*2^A003056(n). - Daniele Parisse, Feb 06 2001

a(n) = 1 + (n - 1 - A003056(n)*(A003056(n) - 1)/2)*2^A003056(n). - Daniele Parisse, Jul 07 2007

Extensions

Edited, corrected and extended by M. F. Hasler, Feb 06 2008

Further edits by N. J. A. Sloane, Feb 08 2008

Upper bound updated with a reference by Max Alekseyev, Nov 23 2008