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Search: id:A004001
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| A004001 |
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Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1))+a(n-a(n-1)) with a(1) = a(2) = 1.
(Formerly M0276)
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+0
79
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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a(n)-a(n-1)=0 or 1 (see the D. Newman reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 06 2005
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REFERENCES
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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.
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.
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
R. K. Guy, Unsolved Problems Number Theory, Sect. E31.
D. R. Hofstadter, personal communication.
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.
C. A. Pickover, Wonders of Numbers, "Cards,Frogs and Fractal sequences" Chapter 96 pp. 217-221 Oxford Univ.Press NY 2000.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
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.
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LINKS
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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.
Experimental Mathematics, Home Page
Nick Hobson, Python program for this sequence
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for Hofstadter-type sequences
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FORMULA
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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 (benoit7848c(AT)orange.fr), Aug 04 2002
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EXAMPLE
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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
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MAPLE
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A004001 := proc(n) option remember; if n<=2 then 1 else A004001(A004001(n-1))+A004001(n-A004001(n-1)); fi; end;
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MATHEMATICA
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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}] (from Robert G. Wilson v)
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CROSSREFS
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Cf. A005229, A005185, A080677, A088359, A093879 (first differences).
Cf. A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Adjacent sequences: A003998 A003999 A004000 this_sequence A004002 A004003 A004004
Sequence in context: A092038 A046699 A102548 this_sequence A086841 A076502 A076897
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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