Hofstadter sequence


Hofstadter sequence

In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.

equences presented in "Gödel, Escher, Bach: an Eternal Golden Braid"

The first Hofstadter sequences were described by Douglas Robert Hofstadter in his book "Gödel, Escher, Bach". In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are:

Hofstadter Figure-Figure sequences

The Hofstadter Figure-Figure (R and S) sequences are defined as follows [Hofstadter (1980) p73] [mathworld|urlname = HofstadterFigure-FigureSequence |title = Hofstadter Figure-Figure Sequence]

:egin{align}R(1)&=1 ; S(1)=2 \R(n)&=R(n-1)+S(n-1), quad n>1.end{align}

with the sequence {S("n")} defined as the positive integers not present in {R("n")}. The first few terms of these sequences are

:R: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, ... OEIS|id=A005228:S: 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, ... OEIS|id=A030124

Hofstadter G sequence

The Hofstadter G sequence is defined as followsHofstadter (1980) p137] [mathworld|urlname = HofstadterG-Sequence |title = Hofstadter G-Sequence]

:egin{align}G(0)&=0 \G(n)&=n-G(G(n-1)), quad n>0.end{align}

The first few terms of this sequence are

:0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, ... OEIS|id=A005206

Hofstadter H sequence

The Hofstadter H sequence is defined as follows [mathworld|urlname = HofstadterH-Sequence |title = Hofstadter H-Sequence]

:egin{align}H(0)&=0 \H(n)&=n-H(H(H(n-1))), quad n>0.end{align}

The first few terms of this sequence are

:0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, ... OEIS|id=A005374

Hofstadter Female and Male sequences

The Hofstadter Female (F) and Male (M) sequences are defined as follows [mathworld|urlname = HofstadterMale-FemaleSequences |title = Hofstadter Male-Female Sequences]

:egin{align}F(0)&=1 ; M(0)=0 \F(n)&=n-M(F(n-1)), quad n>0 \M(n)&=n-F(M(n-1)), quad n>0.end{align}

The first few terms of these sequences are

:F: 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, ... OEIS|id=A005378:M: 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, ... OEIS|id=A005379

Hofstadter Q sequence

The Hofstadter Q sequence is defined as follows [mathworld|urlname = HofstadtersQ-Sequence |title = Hofstadter's Q-Sequence]

:egin{align}Q(1)&=Q(2)=1, \Q(n)&=Q(n-Q(n-1))+Q(n-Q(n-2)), quad n>2.end{align}

The first few terms of the sequence are

:1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, ... OEIS|id=A005185

Hofstadter named the terms of the sequence "Q numbers"; thus the Q number of 6 is 4. The presentation of the Q sequence in Hofstadter's book is actually the first known mentioning of a meta-Fibonacci sequence in literature. [Emerson (2006) p1, p7]

While the terms of the Fibonacci sequence are determined by summing the two preceding terms, the two preceding terms of a Q number determine how far to go back in the Q sequence to find the two terms to be summed. The indexes of the summation terms thus depend on the Q sequence itself.

Q(1), the first element of the sequence (the first Q number) is never the term of any summation in the course of calculating later elements of the sequence; its only use is to provide an index to refer to the second element of the sequence. [Pinn (1999) pp5-6]

Although the terms of the Q sequence seem to flow chaoticPinn (1999) p3] [Pinn (2000) p1] [Emerson (2006) p7] , like many meta-Fibonacci sequences its terms can be grouped into blocks of successive generations. [Pinn (1999) pp3-4] [Balamohan et al. (2007) p19] In case of the Q sequence, the "k"-th generation has 2"k" members. [Pinn (1999) Abstract, p8] Furthermore, with "g" being the generation that a Q number belongs to, the two terms to be summed to calculate the Q number, called its parents, reside by far mostly in generation ("g"-1) and only a few in generation ("g"-2), but never in an even older generation. [Pinn (1999) pp4-5]

Most of these findings are empirical observations since virtually nothing has been proved rigorously about the Q sequence so far.Pinn (1999) p2] [Pinn (2000) p3] Balamohan et al. (2007) p2]

It is specifically unknown if the sequence is well-defined for all "n", that is, if the sequence "dies" at some point because its generation rule tries to refer to terms which would conceptually sit left of the first term Q(1). [Emerson (2006) p7]

Generalizations of the Q sequence

Hofstadter-Huber Qr,s(n) family

20 years after Hofstadter first described the Q sequence, he and Greg Huber used the character Q to name the generalization of the Q sequence towards a family of sequences, and renamed the original Q sequence of his book to U sequence.

The original Q sequence is generalized by replacing ("n"-1) and ("n"-2) by ("n"-"r") and ("n"-"s"), respectively.

This leads to the sequence family

:Q_{r,s}(n) = egin{cases}1 , quad 1 le n le s, \Q_{r,s}(n-Q_{r,s}(n-r))+Q_{r,s}(n-Q_{r,s}(n-s)), quad n > s,end{cases}

where s≥2 and r<s.

With ("r","s") = (1,2), the original Q sequence is a member of this family. So far, only three sequences of the family Qr,s are known, namely the U sequence with ("r","s") = (1,2) (which is the original Q sequence); the V sequence with (r,s) = (1,4); [Balamohan et al. (2007) full article] and the W sequence with (r,s) = (2,4). Only the V sequence, which does not behave as chaotically as the others, is proven not to "die". Similar to the original Q sequence, virtually nothing has been proved rigorously about the W sequence to date.

The first few terms of the V sequence are

:1, 1, 1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 11, ... OEIS|id=A063882

The first few terms of the W sequence are

:1, 1, 1, 1, 2, 4, 6, 7, 7, 5, 3, 8, 9, 11, 12, 9, 9, 13, 11, 9, ... OEIS|id=A087777

For other values ("r","s") the sequences sooner or later "die" i.e. there exists an "n" for which "Qr,s(n)" is undefined because "n-Qr,s(n-r) < 1".

Pinn Fi,j(n) family

In 1998, Klaus Pinn, scientist at University of Münster (Germany) and in close communication with Hofstadter, suggested another generalization of Hofstadter's Q sequence which Pinn called F sequences.Pinn (2000) p16]

The family of Pinn Fi,j sequences is defined as follows:

:F_{i,j}(n) = egin{cases}1 , quad n=1,2, \F_{i,j}(n-i-F_{i,j}(n-1))+F_{i,j}(n-j-F_{i,j}(n-2)), quad n > 2.end{cases}

Thus Pinn introduced additional constants "i" and "j" which shift the index of the terms of the summation conceptually to the left (that is, closer to start of the sequence).

Only F sequences with ("i","j") = (0,0), (0,1), (1,0), and (1,1), the first of which represents the original Q sequence, appear to be well-defined. Unlike Q(1), the first elements of the Pinn Fi,j(n) sequences are terms of summations in calculating later elements of the sequences when any of the additional constants is 1.

The first few terms of the Pinn F0,1 sequence are

:1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 9, ... OEIS|id=A055748

Hofstadter-Conway $10,000 sequence

The Hofstadter-Conway $10,000 sequence is defined as follows [mathworld|urlname = Hofstadter-Conway10000-DollarSequence |title = Hofstadter-Conway $10,000 Sequence]

:egin{align}a(1)&=a(2)=1, \a(n)&=a(a(n-1))+a(n-a(n-1)), quad n>2.end{align}

The first few terms of this sequence are

:1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, ... OEIS|id=A004001

This sequence acquired its name because John Horton Conway offered a prize of $10,000 to anyone who could demonstrate a particular result about its asymptotic behaviour. The prize, subsequently reduced to $1,000, was claimed by Collin Mallows. [ [http://el.media.mit.edu/logo-foundation/pubs/papers/easy_as_11223.html Easy as 1 1 2 2 3] ; Michael Tempel] In private communication with Klaus Pinn, Hofstadter later pointed out he had found the sequence and its structure some 10-15 years before Conway posed his challenge.

Notes

References

*Citation
last1 = Balamohan
first1 = B.
last2 = Kuznetsov
first2 = A.
last3 = Tanny
first3 = Stephan M.
title = On the Behaviour of a Variant of Hofstadter's Q-Sequence
journal = Journal of Integer Sequences
volume = 10
issue = 7
publisher = University of Waterloo
location = Waterloo, Ontario (Canada)
date = 2007-06-27
url = http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Tanny/tanny3.pdf
issn = 1530-7638
.
*Citation
last1 = Emerson
first1 = Nathanial D.
title = A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions
journal = Journal of Integer Sequences
volume = 9
issue = 1
publisher = University of Waterloo
location = Waterloo, Ontario (Canada)
date = 2006-03-17
url = http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Emerson/emerson6.pdf
issn = 1530-7638
.
*citation
last = Hofstadter
first = Douglas
authorlink = Douglas Hofstadter
title = Gödel, Escher, Bach: an Eternal Golden Braid
publisher = Penguin Books
date = 1980
isbn = 0140055797
.
*Citation
last1 = Pinn
first1 = Klaus
contribution = Order and Chaos in Hofstadter's Q(n) Sequence
journal = Complexity
volume = 4
pages = 41–46
year = 1999
id = arxiv|chao-dyn|9803012v2
.
*Citation
last1 = Pinn
first1 = Klaus
contribution = A Chaotic Cousin of Conway's Recursive Sequence
journal = Experiment. Math.
volume = 9
issue = 1
pages = 55–66
year = 2000
id = arxiv|cond-mat|9808031
.


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