|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''Hofstadter sequence''' is a member of a family of related integer sequences defined by [[non-linear system|non-linear]] [[recurrence relations]].
| | Golda is what's written on my beginning certificate although it is not the name on my birth certification. My working day occupation is a travel agent. Playing badminton is a thing that he is totally addicted to. Alaska is exactly where he's usually been living.<br><br>Here is my web page :: tarot readings ([http://conniecolin.com/xe/community/24580 conniecolin.com]) |
| | |
| ==Sequences presented in ''Gödel, Escher, Bach: an Eternal Golden Braid''==
| |
| The first Hofstadter sequences were described by [[Douglas Hofstadter|Douglas Richard 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 a pair of [[Lambek–Moser theorem|complementary integer sequences]] defined as follows<ref>Hofstadter (1980) p 73</ref><ref>{{mathworld|urlname = HofstadterFigure-FigureSequence |title = Hofstadter Figure-Figure Sequence}}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| R(1)&=1~ ;\ S(1)=2 \\
| |
| R(n)&=R(n-1)+S(n-1), \quad n>1.
| |
| \end{align}
| |
| </math>
| |
| | |
| 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 follows<ref name=hof137>Hofstadter (1980) p 137</ref><ref>{{mathworld|urlname = HofstadterG-Sequence |title = Hofstadter G-Sequence}}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| G(0)&=0 \\
| |
| G(n)&=n-G(G(n-1)), \quad n>0.
| |
| \end{align}
| |
| </math>
| |
| | |
| 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<ref name=hof137 /><ref>{{mathworld|urlname = HofstadterH-Sequence |title = Hofstadter H-Sequence}}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| H(0)&=0 \\
| |
| H(n)&=n-H(H(H(n-1))), \quad n>0.
| |
| \end{align}
| |
| </math>
| |
| | |
| 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<ref name=hof137 /><ref>{{mathworld|urlname = HofstadterMale-FemaleSequences |title = Hofstadter Male-Female Sequences}}</ref>
| |
| | |
| :<math>
| |
| \begin{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}
| |
| </math>
| |
| | |
| 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<ref name=hof137 /><ref>{{mathworld|urlname = HofstadtersQ-Sequence |title = Hofstadter's Q-Sequence}}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| Q(1)&=Q(2)=1, \\
| |
| Q(n)&=Q(n-Q(n-1))+Q(n-Q(n-2)), \quad n>2.
| |
| \end{align}
| |
| </math>
| |
| | |
| 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";<ref name=hof137 /> thus the Q number of 6 is 4. The presentation of the Q sequence in Hofstadter's book is actually the first known mention of a [[meta-Fibonacci sequence]] in literature.<ref>Emerson (2006) p 1, p 7</ref>
| |
| | |
| 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 indices of the summation terms thus depend on the Q sequence itself.
| |
| | |
| Q(1), the first element of the sequence, is never one of the two terms being added to produce a later element; it is involved only within an index in the calculation of Q(3).<ref>Pinn (1999) pp 5–6</ref>
| |
| | |
| Although the terms of the Q sequence seem to flow chaotically,<ref name=hof137 /><ref name=pin3>Pinn (1999) p 3</ref><ref>Pinn (2000) p 1</ref><ref name="Emerson 2006 p 7">Emerson (2006) p 7</ref> like many meta-Fibonacci sequences its terms can be grouped into blocks of successive generations.<ref>Pinn (1999) pp 3–4</ref><ref>Balamohan et al. (2007) p 19</ref> In case of the Q sequence, the ''k''-th generation has 2<sup>''k''</sup> members.<ref>Pinn (1999) Abstract, p 8</ref> 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.<ref>Pinn (1999) pp 4–5</ref>
| |
| | |
| Most of these findings are empirical observations, since virtually nothing has been proved rigorously about the ''Q'' sequence so far.<ref name=pin2>Pinn (1999) p 2</ref><ref>Pinn (2000) p 3</ref><ref name=bal2>Balamohan et al. (2007) p 2</ref> 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).<ref name="Emerson 2006 p 7"/><ref name=pin2 /><ref name=bal2 />
| |
| | |
| ==Generalizations of the ''Q'' sequence==
| |
| ===Hofstadter–Huber ''Q''<sub>''r'',''s''</sub>(''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.<ref name=bal2 />
| |
| | |
| The original ''Q'' sequence is generalized by replacing (''n'' − 1) and (''n'' − 2) by (''n'' − ''r'') and (''n'' − ''s''), respectively.<ref name=bal2 />
| |
| | |
| This leads to the sequence family
| |
| | |
| :<math>
| |
| Q_{r,s}(n) =
| |
| \begin{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}
| |
| </math>
| |
| | |
| 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 ''Q''<sub>''r'',''s''</sub> are known, namely the ''U'' sequence with (''r'',''s'') = (1,2) (which is the original ''Q'' sequence);<ref name=bal2 /> the ''V'' sequence with (''r'',''s'') = (1,4);<ref>Balamohan et al. (2007) full article</ref> and the W sequence with (r,s) = (2,4).<ref name=bal2 /> Only the V sequence, which does not behave as chaotically as the others, is proven not to "die".<ref name=bal2 /> Similar to the original ''Q'' sequence, virtually nothing has been proved rigorously about the W sequence to date.<ref name=bal2 />
| |
| | |
| 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 ''Q<sub>''r'',''s''</sub>(''n'') is undefined because ''n'' − ''Q''<sub>''r'',''s''</sub>(''n'' − ''r'') < 1.<ref name=bal2 />
| |
| | |
| ===Pinn ''F''<sub>''i'',''j''</sub>(''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.<ref name=pin16>Pinn (2000) p 16</ref>
| |
| | |
| The family of Pinn ''F''<sub>''i'',''j''</sub> sequences is defined as follows:
| |
| | |
| :<math> | |
| F_{i,j}(n) =
| |
| \begin{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}
| |
| </math>
| |
| | |
| 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).<ref name=pin16 />
| |
| | |
| 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.<ref name=pin16 /> Unlike ''Q''(1), the first elements of the Pinn ''F''<sub>''i'',''j''</sub>(''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 ''F''<sub>0,1</sub> 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==
| |
| <!-- Image with unknown copyright status removed: [[File:conway_challenge_sequence.gif|thumb|300px|right|A plot of a(n)/n, which tends to 1/2 as proved by [[Conway]]]] -->
| |
| The Hofstadter–Conway $10,000 sequence is defined as follows<ref>{{mathworld|urlname = Hofstadter-Conway10000-DollarSequence |title = Hofstadter-Conway $10,000 Sequence}}</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| a(1)&=a(2)=1, \\
| |
| a(n)&=a(a(n-1))+a(n-a(n-1)), \quad n>2.
| |
| \end{align}
| |
| </math>
| |
| | |
| 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 analysis|asymptotic]] behaviour. The prize, since reduced to $1,000, was claimed by [[Collin Mallows]].<ref>[http://el.media.mit.edu/logo-foundation/pubs/papers/easy_as_11223.html Easy as 1 1 2 2 3]; Michael Tempel</ref> In private communication with [[Klaus Pinn]], Hofstadter later claimed he had found the sequence and its structure some 10–15 years before Conway posed his challenge.<ref name=pin3 />
| |
| | |
| ==Notes==
| |
| {{Reflist|2}}
| |
| | |
| ==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
| |
| |format=PDF| issn = 1530-7638 }}.
| |
| *{{Citation
| |
| | last1 = Emerson
| |
| | first1 = Nathaniel 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
| |
| |format=PDF| issn = 1530-7638 }}.
| |
| *{{citation
| |
| | last = Hofstadter
| |
| | first = Douglas
| |
| | authorlink = Douglas Hofstadter
| |
| | title = Gödel, Escher, Bach: an Eternal Golden Braid
| |
| | publisher = Penguin Books
| |
| | year = 1980
| |
| | isbn = 0-14-005579-7}}.
| |
| *{{Citation
| |
| | last1 = Pinn
| |
| | first1 = Klaus
| |
| | contribution = Order and Chaos in Hofstadter's Q(n) Sequence
| |
| | journal = Complexity
| |
| | volume = 4
| |
| | issue = 3
| |
| | pages = 41–46
| |
| | year = 1999
| |
| | arxiv = chao-dyn/9803012v2 | doi = 10.1002/(SICI)1099-0526(199901/02)4:3<41::AID-CPLX8>3.0.CO;2-3
| |
| | title = Order and chaos in Hofstadter'sQ(n) sequence }}.
| |
| *{{Citation
| |
| | last1 = Pinn
| |
| | first1 = Klaus
| |
| | contribution = A Chaotic Cousin of Conway's Recursive Sequence
| |
| | journal = [[Experimental Mathematics (journal)|Experimental Mathematics]]
| |
| | volume = 9
| |
| | issue = 1
| |
| | pages = 55–66
| |
| | year = 2000
| |
| | arxiv = cond-mat/9808031 }}.
| |
| | |
| [[Category:Integer sequences]]
| |