

A005185


Hofstadter Qsequence: a(1) = a(2) = 1; a(n) = a(na(n1)) + a(na(n2)) for n > 2.
(Formerly M0438)


93



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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(A081829(n)+1) < a(A081829(n)); a(A081828(n)+1) = a(A081828(n)); a(A081830(n)+1) > a(A081830(n)); a(A194626(n)+1) = a(A194626(n)) + 1.  Reinhard Zumkeller, Sep 15 2011


REFERENCES

B. W. Conolly, "MetaFibonacci sequences," in S. Vajda, editor, Fibonacci and Lucas Numbers and the Golden Section. Halstead Press, NY, 1989, pp. 127138.
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
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Qsequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
P. Bourke, Hofstadter "Q" Series
P. Bourke, Hofstadter "Q" Series [Cached copy, pdf only, with permission]
P. 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 metaFibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794824. 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
Nathaniel D. Emerson, A Family of MetaFibonacci Sequences Defined by VariableOrder Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]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), 149161.
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186190; 94 (1987), 965; 96 (1989), 905.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, EtaLore [Cached copy, with permission]
D. R. Hofstadter, PiMu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
D. R. Hofstadter, Curious patterns and nonpatterns in a family of metaFibonacci 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 nonhomogeneous nested recursions using trees, Annals of Combinatorics 17.4 (2013): 695710. See p. 695.  N. J. A. Sloane, Apr 16 2014
K. Pinn, Order and chaos in Hofstadter's Q(n) sequence, Complexity, 4:3 (1999), 4146.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 5565.
K. Pinn, A Chaotic Cousin Of Conway's Recursive Sequence
Eric Weisstein's World of Mathematics, Hofstadter's QSequence
Wikipedia, Hofstadter sequence
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadtertype sequences


EXAMPLE

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.


MAPLE

A005185 := proc(n) option remember;
if n<=2 then
1
elif n > procname(n1) and n > procname(n2) then
RETURN(procname(nprocname(n1))+procname(nprocname(n2)));
else
ERROR(" died at n= ", n);
fi;
end proc;
# More generally, the following defines the HofstadterHuber 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(nr) <= n) and (a(ns) <= n) then
a(na(nr))+a(na(ns));
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, 1, 70} ]


PROG

(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)))))))))
(Mupad) q:=proc(n) option remember; begin if n<=2 then 1 else q(nq(n1))+q(nq(n2)) 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[kA[k1]]+ A[kA[k2]]); A[n])} /* Michael Somos, Jul 16 2007 */
(Haskell)
a005185 n = a005185_list !! (n1)
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
(C)
#include <stdio.h>
#define LIM 20
int Qa[LIM];
int Q(int n){if (n==1  n==2){return 1; } else{return Qa[nQa[n1]]+Qa[nQa[n2]]; }}
int main(){int i; printf("n\tQ\n"); for(i=1; i<LIM; i+=1){printf("%d\t%d\n", i, Q(i)); Qa[i]=Q(i); }printf("\n"); } // Gonzalo Ciruelos, Aug 01 2013
(MAGMA) I:=[1, 1]; [n le 2 select I[n] else Self(nSelf(n1))+Self(nSelf(n2)): n in [1..90]]; // Vincenzo Librandi, Aug 08 2014


CROSSREFS

Cf. A004001, A005206, A005374, A005375, A005378, A005379, A070867, A226222, A239913.
Cf. A081827 (first differences).
Cf. A226244, A226245 (record values and where they occur).
See A244477 for a different start.
Sequence in context: A080595 A123579 A166493 * A119466 A100922 A047785
Adjacent sequences: A005182 A005183 A005184 * A005186 A005187 A005188


KEYWORD

nonn,nice,easy,look


AUTHOR

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


STATUS

approved



