A005229 a(1) = a(2) = 1; for n > 2, a(n) = a(a(n-2)) + a(n - a(n-2)). (Formerly M0441)

1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 30, 30, 31, 32, 33, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 45, 45, 46

Offset

1,3

Comments

By induction a(n) <= n, but an exact rate of growth is not known.

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018), Article ID 8517125.

Nick Hobson, Python program for this sequence

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

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

Eric Weisstein's World of Mathematics, Mallows' Sequence.

Maple

A005229:= proc(n) option remember;
     if n<=2 then 1 else A005229(A005229(n-2)) +A005229(n-A005229(n-2));
     fi; end;
seq(A005229(n), n=1..70)

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n-2]] + a[n - a[n-2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 06 2013 *)

Prog

(PARI) a(n)=an[n]; an=vector(100, n, 1); for(n=3, 100, an[n]=a(a(n-2))+a(n-a(n-2)))
(Haskell)
import Data.Function (on)
a005229 n = a005229_list !! (n-1)
a005229_list = 1 : 1 : zipWith ((+) `on` a005229)
                       a005229_list (zipWith (-) [3..] a005229_list)
-- Reinhard Zumkeller, Jan 17 2014
(Sage)
@CachedFunction
def a(n): # A005229
    if (n<3): return 1
    else: return a(a(n-2)) + a(n-a(n-2))
[a(n) for n in (1..100)] # G. C. Greubel, Mar 27 2022

Crossrefs

Cf. A004001, A051105, A087758, A116591, A147880, A169638.

Sequence in context: A132351 A025556 A338006 * A091245 A347649 A368908

Adjacent sequences: A005226 A005227 A005228 * A005230 A005231 A005232

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Typo in definition corrected by Nick Hobson, Feb 21 2007

Status

approved