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

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice
• easy

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.

Crossrefs

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

Programs

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

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 *)

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)))

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

Extensions

Typo in definition corrected by Nick Hobson, Feb 21 2007