A003160 a(1) = a(2) = 1, a(n) = n - a(a(n-1)) - a(a(n-2)).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Sequence of indices n where a(n-1) < a(n) appears to be given by A003156. - Joerg Arndt, May 11 2010

The number n appears A080426(n+1) times. - John Keith, Dec 31 2020

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

Crossrefs

Cf. A003156, A005206, A080426, A095774, A095775.

Programs

Haskell

a003160 n = a003160_list !! (n-1)
a003160_list = 1 : 1 : zipWith (-) [3..] (zipWith (+) xs $ tail xs)
   where xs = map a003160 a003160_list
-- Reinhard Zumkeller, Aug 02 2013

Mathematica

Block[{a = {1, 1}}, Do[AppendTo[a, i - a[[ a[[-1]] ]] - a[[ a[[-2]] ]] ], {i, 3, 76}]; a] (* Michael De Vlieger, Dec 31 2020 *)

PARI

a(n)=if(n<3,1,n-a(a(n-1))-a(a(n-2)))

SageMath

@CachedFunction
def a(n): return 1 if (n<3) else n - a(a(n-1)) - a(a(n-2))
[a(n) for n in range(1, 81)] # G. C. Greubel, Nov 06 2022

Formula

a(n) is asymptotic to n/2.

Conjecture: a(n) = E/2 where we start with A := n + 1, B := 0, L := A085423(A), C := A000975(L-1), D := 0, E := C and until A = B consecutively apply B := A, A := 2*C - A - (L mod 2) + 2, L := A085423(A), C := A000975(L-1), D := D + 1, E := (1 + [A = B])*E + (-1)^D*C. - Mikhail Kurkov, May 12 2025

Extensions

Edited by Benoit Cloitre, Jan 01 2003