login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.
(Formerly M2449)
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

It is conjectured that the algorithm always gives the optimal solution; for n<=30 this is confirmed by exhaustive search, but no proof is known for the general case.

"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

REFERENCES

A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math., 8 (1972), 169-176.

Cull, Paul; Ecklund, E. F. 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, Apr 08 2012

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013.

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

M. F. Hasler and 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. M. Hinz, An iterative algorithm for the Tower of Hanoi with four pegs

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013.

B. Houston and H. Masum, Explorations in 4-peg Tower of Hanoi [Ppaper]

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.

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(2^A003056(i), i=0..n-1). - Daniele Parisse (daniele.parisse(AT)m.eads.net), May 09 2003

a(n) = 1 + (n + A003056(n) - 1 - A003056(n)*(A003056(n) + 1)/2)*2^A003056(n). - Daniele Parisse (daniele.parisse(AT)m.dasa.de), Feb 06 2001

a(n) = 1 + (n - 1 - A003056(n)*(A003056(n) - 1)/2)*2^A003056(n). - Daniele Parisse (daniele.parisse(AT)t-online.de), Jul 07 2007

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 *)

PROG

(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

(Haskell)

a007664 = sum . map (a000079 . a003056) . enumFromTo 0 . subtract 1

-- Reinhard Zumkeller, Feb 17 2013

CROSSREFS

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

Sequence in context: A061571 A049690 A080075 * A215812 A114395 A075314

Adjacent sequences:  A007661 A007662 A007663 * A007665 A007666 A007667

KEYWORD

nonn,nice

AUTHOR

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

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 21 08:37 EDT 2017. Contains 289638 sequences.