|
| |
|
|
A004001
|
|
Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1))+a(n-a(n-1)) with a(1) = a(2) = 1.
(Formerly M0276)
|
|
117
|
|
|
|
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
|
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.
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.
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.
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, 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}] (from 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
|
|
|
CROSSREFS
|
Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences).
Cf. A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Sequence in context: A046699 A218446 A102548 * A086841 A076502 A076897
Adjacent sequences: A003998 A003999 A004000 * A004002 A004003 A004004
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
