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A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1))+a(n-a(n-1)) with a(1) = a(2) = 1.
(Formerly M0276)
139
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 July 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(n-1)=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

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), 85-94.

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 Number Theory, Sect. E31.

D. R. Hofstadter, personal communication.

C. A. Pickover, Wonders of Numbers, "Cards,Frogs and Fractal sequences" Chapter 96 pp. 217-221 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

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.

F. Michel Dekking, Pure morphic sequences and their standard forms, arXiv:1509.00260 [math.CO], 2015.

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.

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.

Nick Hobson, Python program for this sequence

D. R. Hofstadter, Plot of 100000 terms of a(n)-1/2

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, October 10 2014; Part 1, Part 2.

D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.

T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.

D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.

John A. Pelesko, Generalizing the Conway-Hofstadter $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), 55-65.

N. J. A. Sloane, Scan of notebook pages with notes on John Conway's Colloquium talk on July 15, 1988 [See Comments above]

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Eric Weisstein's World of Mathematics, Hofstadter-Conway 10000-Dollar Sequence.

Eric Weisstein's World of Mathematics, Newman-Conway Sequence

Wikipedia, Hofstadter sequence

Index entries for Hofstadter-type sequences

FORMULA

lim n ->infinity a(n)/n = 1/2 and as special cases, if n>0, a(2^n-i) = 2^(n-1) for 0<=i<=n-1; a(2^n-1)=2^(n-1)-1; a(2^n+1)=2^(n-1)+1. - Benoit Cloitre, Aug 04 2002

EXAMPLE

If n=4 2^4=16, a(16-i)=2^(4-1)=8 for 0<=i<=4-1=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(n-1)) +procname(n-procname(n-1)); 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 !! (n-1)

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[n-1]]+a[n-a[n-1]]); a \\ Charles R Greathouse IV, Jun 10 2011

(Scheme)

;; An implementation of memoization-macro 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[c-1]] + A[c-A[c-1]]

        c += 1

    return A[n]

# Edward Minnix III, Nov 02 2015

CROSSREFS

Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences).

Cf. A004074 (A249071), A005350, A005707, A093878. Different from A086841. Run lengths give A051135.

Sequence in context: A239105 A218446 A102548 * A086841 A076502 A076897

Adjacent sequences:  A003998 A003999 A004000 * A004002 A004003 A004004

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 27 23:55 EDT 2016. Contains 273424 sequences.