A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, 28, 26, 30, 30, 28, 32, 30, 32, 32, 32, 32, 40, 33, 31, 38, 35, 33, 39, 40, 37, 38, 40, 39

Offset: 1

Author

Simon Plouffe and N. J. A. Sloane, May 20 1991

Keywords

• nonn
• nice
• look

Comments

Rate of growth is not known. In fact it is not even known if this sequence is defined for all positive n.

Roman Pearce, Aug 29 2014, has computed that a(n) exists for n <= 10^10. - N. J. A. Sloane

a(n) exists for n <= 3*10^10. - M. Eric Carr, Jul 02 2023

Examples

			a(18) = 11 because a(17) is 10 and a(16) is 9, so we take a(18 - 10) + a(18 - 9) = a(8) + a(9) = 5 + 6 = 11.
		

References

• B. W. Conolly, "Meta-Fibonacci sequences," in S. Vajda, editor, Fibonacci and Lucas Numbers and the Golden Section. Halstead Press, NY, 1989, pp. 127-138.
• R. K. Guy, Unsolved Problems in Number Theory, Sect. E31.
• D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 138.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Links

• T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
• Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics, Vol. 16 (2018), pp. 1490-1500.
• Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
• Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, 2017.
• B. Balamohan, A. Kuznetsov, and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
• Thomas Bloom, Problem 422, Erdős Problems.
• Paul Bourke, Hofstadter "Q" Series.
• Paul Bourke, Hofstadter "Q" Series. [Cached copy, pdf only, with permission]
• Paul Bourke, Listen to first 300 terms of this sequence. [Cached copy from the Hofstadter "Q" web page, with permission]
• Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See p. 794.
• Benoit Cloitre, Plot of a(n)/n - 1/2 for n=1 up to 2^19.
• J.-P. Davalan, Douglas Hofstadter's sequences.
• Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023.
• Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
• Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
• Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
• Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
• Nathan Fox, A New Approach to the Hofstadter Q-Recurrence, arXiv:1807.01365 [math.NT], 2018.
• S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)].
• J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
• R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
• R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
• Nick Hobson, Python program for this sequence.
• D. R. Hofstadter, Eta-Lore. [Cached copy, with permission]
• D. R. Hofstadter, Pi-Mu Sequences. [Cached copy, with permission]
• D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991.
• D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
• D. R. Hofstadter, Plot of first 100 million terms.
• Abraham Isgur, Mustazee Rahman, and Stephen Tanny, Solving non-homogeneous nested recursions using trees, Annals of Combinatorics 17.4 (2013): 695-710. See p. 695. - _N. J. A. Sloane_, Apr 16 2014
• A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505.
• Piotr Miska, Bartosz Sobolewski, and Maciej Ulas, Binary sequences meet the Fibonacci sequence, arXiv:2412.11319 [math.NT], 2024. See p. 20.
• K. Pinn, Order and chaos in Hofstadter's Q(n) sequence, Complexity, 4:3 (1999), 41-46.
• K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
• K. Pinn, A Chaotic Cousin Of Conway's Recursive Sequence, arXiv:cond-mat/9808031 [cond-mat.stat-mech], 1998.
• N. J. A. Sloane and Brady Haran, Amazing Graphs III, Numberphile video (2019).
• I. Vardi, Email to N. J. A. Sloane, Jun. 1991.
• A. M. M. Sharif Ullah, Surface Roughness Modeling Using Q-Sequence, Mathematical and Computational Applications, 2017.
• Eric Weisstein's World of Mathematics, Hofstadter's Q-Sequence.
• Wikipedia, Hofstadter sequence.
• Pedro Zanetti, C++ code snippet for this sequence.
• Index entries for sequences from "Goedel, Escher, Bach"
• Index entries for Hofstadter-type sequences

Crossrefs

Cf. A004001, A005206, A005374, A005375, A005378.

Cf. A005379, A070867, A226222, A239913, A284019.

Cf. A081827 (first differences).

Cf. A226244, A226245 (record values and where they occur).

See A244477 for a different start.

Programs

C

#include 
#define LIM 20
int Qa[LIM];
int Q(int n){if (n==1 || n==2){return 1;} else{return Qa[n-Qa[n-1]]+Qa[n-Qa[n-2]];}}
int main(){int i;printf("n\tQ\n");for(i=1; iGonzalo Ciruelos, Aug 01 2013

Haskell

a005185 n = a005185_list !! (n-1)
a005185_list = 1 : 1 : zipWith (+)
   (map a005185 $ zipWith (-) [3..] a005185_list)
   (map a005185 $ zipWith (-) [3..] $ tail a005185_list)
-- Reinhard Zumkeller, Jun 02 2013, Sep 15 2011

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-2)): n in [1..90]]; // Vincenzo Librandi, Aug 08 2014

Maple

A005185 := proc(n) option remember;
    if n<=2 then 1
    elif n > procname(n-1) and n > procname(n-2) then
        RETURN(procname(n-procname(n-1))+procname(n-procname(n-2)));
    else
        ERROR(" died at n= ", n);
    fi; end proc;
# More generally, the following defines the Hofstadter-Huber sequence Q(r,s) - N. J. A. Sloane, Apr 15 2014
r:=1; s:=2;
a:=proc(n) option remember; global r,s;
if n <= s then  1
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi; fi; end;
[seq(a(n), n=1..100)];

Mathematica

a[1]=a[2]=1; a[n_]:= a[n]= a[n -a[n-1]] + a[n -a[n-2]]; Table[ a[n], {n,70}]

MuPAD

q:=proc(n) option remember; begin if n<=2 then 1 else q(n-q(n-1))+q(n-q(n-2)) end_if; end_proc: q(i)$i=1..100; // Zerinvary Lajos, Apr 03 2007

PARI

{a(n)= local(A); if(n<1, 0, A=vector(n,k,1); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A[n])} /* Michael Somos, Jul 16 2007 */

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n):
    if n < 3: return 1
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 26 2021

Sage

@CachedFunction
def a(n):
    if (n<3): return 1
    else: return a(n -a(n-1)) + a(n -a(n-2))
[a(n) for n in (1..70)] # G. C. Greubel, Feb 13 2020

Scheme

(define q (lambda (n) (cond ( (eqv? n 0) 1) ( (eqv? n 1) 1) ( #t (+ (q (- n (q (- n 1)))) (q (- n (q (- n 2)))))))))

Scheme

;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A005185 n) (if (<= n 2) 1 (+ (A005185 (- n (A005185 (- n 1)))) (A005185 (- n (A005185 (- n 2)))))))
;; Antti Karttunen, Mar 22 2017