A003160 - OEIS

%I M0446 #43 Jan 05 2025 19:51:33

%S 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,

%T 16,16,16,17,17,17,18,19,20,20,20,21,21,21,22,22,22,23,24,25,25,25,26,

%U 26,26,27,27,27,28,29,30,30,30,31,32,33,33,33,34,35,36,36,36,37,37,37,38

%N a(1) = a(2) = 1, a(n) = n - a(a(n-1)) - a(a(n-2)).

%C Sequence of indices n where a(n-1) < a(n) appears to be given by A003156. - _Joerg Arndt_, May 11 2010

%C The number n appears A080426(n+1) times. - _John Keith_, Dec 31 2020

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

%H Reinhard Zumkeller, <a href="/A003160/b003160.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

%F a(n) is asymptotic to n/2.

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

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

%o (Haskell)

%o a003160 n = a003160_list !! (n-1)

%o a003160_list = 1 : 1 : zipWith (-) [3..] (zipWith (+) xs $ tail xs)

%o    where xs = map a003160 a003160_list

%o -- _Reinhard Zumkeller_, Aug 02 2013

%o (SageMath)

%o @CachedFunction

%o def a(n): return 1 if (n<3) else n - a(a(n-1)) - a(a(n-2))

%o [a(n) for n in range(1, 81)] # _G. C. Greubel_, Nov 06 2022

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

%K nonn

%O 1,4

%A _N. J. A. Sloane_

%E Edited by _Benoit Cloitre_, Jan 01 2003