

A004001


HofstadterConway $10000 sequence: a(n) = a(a(n1)) + a(na(n1)) with a(1) = a(2) = 1.
(Formerly M0276)


210



1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

On Jul 15 1988 during a colloquium talk at Bell Labs, John Conway stated that he could prove that a(n)/n > 1/2 as n approached infinity, but that the proof was extremely difficult. He therefore offered $100 to someone who could find an n_0 such that for all n >= n_0, we have a(n)/n  1/2 < 0.05, and he offered $10,000 for the least such n_0. I took notes (a scan of my notebook pages appears below), plus the talk  like all Bell Labs Colloquia at that time  was recorded on video. John said afterwards that he meant to say $1000, but in fact he said $10,000. I was in the front row. The prize was claimed by Colin Mallows, who agreed not to cash the check.  N. J. A. Sloane, Oct 21 2015
a(n)  a(n1) = 0 or 1 (see the D. Newman reference).  Emeric Deutsch, Jun 06 2005
a(A188163(n)) = n and a(m) < n for m < A188163(n).  Reinhard Zumkeller, Jun 03 2011
From Daniel Forgues, Oct 04 2019: (Start)
Conjectures:
a(n) = n/2 iff n = 2^k, k >= 1.
a(n) = 2^(k1): k times, for n = 2^k  (k1) to 2^k, k >= 1. (End)


REFERENCES

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 8594.
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 Number Theory, Sect. E31.
D. R. Hofstadter, personal communication.
C. A. Pickover, Wonders of Numbers, "Cards, Frogs and Fractal sequences", Chapter 96, pp. 217221, Oxford Univ. Press, NY, 2000.
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, Table of n, a(n) for n = 1..10000
Altug Alkan, On a Generalization of Hofstadter's QSequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan and O. Ozgur Aybar, On Families of Solutions for MetaFibonacci Recursions Related to HofstadterConway $10000 Sequence, Slides of talk at presented in 5th International Interdisciplinary Chaos Symposium on Chaos and Complex Systems, May 912, 2019.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, Volume 2017, Article ID 2614163, 8 pages.
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.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathaniel D. Emerson, A Family of MetaFibonacci Sequences Defined by VariableOrder Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Nathan Fox, LinearRecurrent Solutions to MetaFibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
Nathan Fox, A Slow Relative of Hofstadter's QSequence, arXiv:1611.08244 [math.NT], 2016.
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, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, Plot of 100000 terms of a(n)n/2
D. R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, Oct 10 2014; Part 1, Part 2.
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, 11281147. (20 pages); DOI:10.1137/15M1040505.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958959.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225252.
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 520.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
John A. Pelesko, Generalizing the ConwayHofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 5565.
N. J. A. Sloane, Scan of notebook pages with notes on John Conway's Colloquium talk on Jul 15 1988 [See Comments above]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991
Eric Weisstein's World of Mathematics, HofstadterConway 10000Dollar Sequence.
Eric Weisstein's World of Mathematics, NewmanConway Sequence
Wikipedia, Hofstadter sequence
Index entries for sequences related to binary expansion of n
Index entries for Hofstadtertype sequences


FORMULA

Limit_{n>infinity} a(n)/n = 1/2 and as special cases, if n > 0, a(2^ni) = 2^(n1) for 0 <= i < = n1; a(2^n+1) = 2^(n1) + 1.  Benoit Cloitre, Aug 04 2002 [Corrected by Altug Alkan, Apr 03 2017]


EXAMPLE

If n=4, 2^4=16, a(16i) = 2^(41) = 8 for 0 <= i <= 41 = 3, hence a(16)=a(15)=a(14)=a(13)=8.


MAPLE

A004001 := proc(n) option remember; if n<=2 then 1 else procname(procname(n1)) +procname(nprocname(n1)); fi; end;


MATHEMATICA

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n  1]] + a[n  a[n  1]]; Table[ a[n], {n, 1, 75}] (* Robert G. Wilson v *)


PROG

(Haskell)
a004001 n = a004001_list !! (n1)
a004001_list = 1 : 1 : h 3 1 { memoization }
where h n x = x' : h (n + 1) x'
where x' = a004001 x + a004001 (n  x)
 Reinhard Zumkeller, Jun 03 2011
(PARI) a=vector(100); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n1]]+a[na[n1]]); a \\ Charles R Greathouse IV, Jun 10 2011
(PARI) first(n)=my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k1]]+v[kv[k1]]); v \\ Charles R Greathouse IV, Feb 26 2017
(Scheme)
;; An implementation of memoizationmacro definec can be found for example from: http://oeis.org/wiki/Memoization
(definec (A004001 n) (if (<= n 2) 1 (+ (A004001 (A004001 ( n 1))) (A004001 ( n (A004001 ( n 1)))))))
;; Antti Karttunen, Oct 22 2014
(Python)
def a004001(n):
A = {1: 1, 2: 1}
c = 1 #counter
while n not in A.keys():
if c not in A.keys():
A[c] = A[A[c1]] + A[cA[c1]]
c += 1
return A[n]
# Edward Minnix III, Nov 02 2015
(MAGMA) I:=[1, 1]; [n le 2 select I[n] else Self(Self(n1))+ Self(nSelf(n1)):n in [1..75]]; // Marius A. Burtea, Aug 16 2019


CROSSREFS

Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences), A265332, A266341, A055748 (a chaotic cousin).
Cf. A004074 (A249071), A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Cf. also permutations A267111A267112 and arrays A265901, A265903.
Sequence in context: A102548 A343174 A340205 * A086841 A076502 A076897
Adjacent sequences: A003998 A003999 A004000 * A004002 A004003 A004004


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane


STATUS

approved



