Hofstadter Sequence

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## Sequences presented in *Gödel, Escher, Bach: an Eternal Golden Braid*

### Hofstadter Figure-Figure sequences

### Hofstadter G sequence

### Hofstadter H sequence

### Hofstadter Female and Male sequences

### Hofstadter Q sequence

## Generalizations of the *Q* sequence

### Hofstadter-Huber *Q*_{r,s}(*n*) family

### Pinn *F*_{i,j}(*n*) family

## Hofstadter-Conway $10,000 sequence

## References

### Sources

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Hofstadter Sequence

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

The first Hofstadter sequences were described by 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:

The Hofstadter Figure-Figure (R and S) sequences are a pair of complementary integer sequences defined as follows^{[1]}^{[2]}

with the sequence defined as a strictly increasing series of positive integers not present in . 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, ... (sequence in the OEIS)
- S: 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, ... (sequence in the OEIS)

The Hofstadter G sequence is defined as follows^{[3]}^{[4]}

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, ... (sequence in the OEIS)

The Hofstadter H sequence is defined as follows^{[3]}^{[5]}

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, ... (sequence in the OEIS)

The Hofstadter Female (F) and Male (M) sequences are defined as follows^{[3]}^{[6]}

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, ... (sequence in the OEIS)
- M: 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, ... (sequence in the OEIS)

The Hofstadter Q sequence is defined as follows^{[3]}^{[7]}

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, ... (sequence in the OEIS)

Hofstadter named the terms of the sequence "Q numbers";^{[3]} 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.^{[8]}

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).^{[9]}

Although the terms of the Q sequence seem to flow chaotically,^{[3]}^{[10]}^{[11]}^{[12]} like many meta-Fibonacci sequences its terms can be grouped into blocks of successive generations.^{[13]}^{[14]} In case of the Q sequence, the *k*-th generation has 2^{k} members.^{[15]} 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.^{[16]}

Most of these findings are empirical observations, since virtually nothing has been proved rigorously about the *Q* sequence so far.^{[17]}^{[18]}^{[19]} 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).^{[12]}^{[17]}^{[19]}

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 toward a family of sequences, and renamed the original *Q* sequence of his book to *U* sequence.^{[19]}

The original *Q* sequence is generalized by replacing (*n* − 1) and (*n* − 2) by (*n* − *r*) and (*n* − *s*), respectively.^{[19]}

This leads to the sequence family

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*_{r,s} are known, namely the *U* sequence with (*r*,*s*) = (1,2) (which is the original *Q* sequence);^{[19]} the *V* sequence with (*r*,*s*) = (1,4);^{[20]} and the W sequence with (r,s) = (2,4).^{[19]} Only the V sequence, which does not behave as chaotically as the others, is proven not to "die".^{[19]} Similar to the original *Q* sequence, virtually nothing has been proved rigorously about the W sequence to date.^{[19]}

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, ... (sequence in the OEIS)

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, ... (sequence in the OEIS)

For other values (*r*,*s*) the sequences sooner or later "die" i.e. there exists an *n* for which *Q*_{r,s}(*n*) is undefined because *n* − *Q*_{r,s}(*n* − *r*) < 1.^{[19]}

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.^{[21]}

The family of Pinn *F*_{i,j} sequences is defined as follows:

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).^{[21]}

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.^{[21]} Unlike *Q*(1), the first elements of the Pinn *F*_{i,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 *F*_{0,1} sequence are

- 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 9, 10, 10, 11, ... (sequence in the OEIS)

The Hofstadter-Conway $10,000 sequence is defined as follows^{[22]}

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, ... (sequence in the OEIS)

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 behavior. The prize, since reduced to $1,000, was claimed by Collin Mallows.^{[23]} 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.^{[10]}

**^**Hofstadter (1980), p. 73**^**Weisstein, Eric W. "Hofstadter Figure-Figure Sequence".*MathWorld*.- ^
^{a}^{b}^{c}^{d}^{e}^{f}Hofstadter (1980), p. 137 **^**Weisstein, Eric W. "Hofstadter G-Sequence".*MathWorld*.**^**Weisstein, Eric W. "Hofstadter H-Sequence".*MathWorld*.**^**Weisstein, Eric W. "Hofstadter Male-Female Sequences".*MathWorld*.**^**Weisstein, Eric W. "Hofstadter's Q-Sequence".*MathWorld*.**^**Emerson (2006), pp. 1, 7**^**Pinn (1999), pp. 5-6- ^
^{a}^{b}Pinn (1999), p. 3 **^**Pinn (2000), p. 1- ^
^{a}^{b}Emerson (2006), p. 7 **^**Pinn (1999), pp. 3-4**^**Balamohan, Kuznetsov & Tanny (2007), p. 19**^**Pinn (1999), Abstract, p. 8**^**Pinn (1999), pp. 4-5- ^
^{a}^{b}Pinn (1999), p. 2 **^**Pinn (2000), p. 3- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Balamohan, Kuznetsov & Tanny (2007), p. 2 **^**Balamohan, Kuznetsov & Tanny (2007), full article- ^
^{a}^{b}^{c}Pinn (2000), p. 16 **^**Weisstein, Eric W. "Hofstadter-Conway $10,000 Sequence".*MathWorld*.**^**Tempel, Michael. "Easy as 1 1 2 2 3" (PDF). Cite journal requires`|journal=`

(help)

- Balamohan, B.; Kuznetsov, A.; Tanny, Stephan M. (2007-06-27), "On the Behaviour of a Variant of Hofstadter's Q-Sequence" (PDF),
*Journal of Integer Sequences*, Waterloo, Ontario (Canada): University of Waterloo,**10**, ISSN 1530-7638. - Emerson, Nathaniel D. (2006-03-17), "A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions" (PDF),
*Journal of Integer Sequences*, Waterloo, Ontario (Canada): University of Waterloo,**9**(1), ISSN 1530-7638. - Hofstadter, Douglas (1980),
*Gödel, Escher, Bach: an Eternal Golden Braid*, Penguin Books, ISBN 0-14-005579-7. - Pinn, Klaus (1999), "Order and Chaos in Hofstadter's Q(n) Sequence",
*Complexity*,**4**(3): 41-46, arXiv:chao-dyn/9803012v2, doi:10.1002/(SICI)1099-0526(199901/02)4:3<41::AID-CPLX8>3.0.CO;2-3. - Pinn, Klaus (2000), "A Chaotic Cousin of Conway's Recursive Sequence",
*Experimental Mathematics*,**9**(1): 55-66, arXiv:cond-mat/9808031, Bibcode:1998cond.mat..8031P.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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