A064437 a(1)=1, a(n) = a(n-1) + 3 if n is already in the sequence, a(n) = a(n-1) + 2 otherwise.

1, 3, 6, 8, 10, 13, 15, 18, 20, 23, 25, 27, 30, 32, 35, 37, 39, 42, 44, 47, 49, 51, 54, 56, 59, 61, 64, 66, 68, 71, 73, 76, 78, 80, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 109, 112, 114, 117, 119, 122, 124, 126, 129, 131, 134, 136, 138, 141, 143, 146, 148, 150

Offset: 1

Author

Benoit Cloitre, Feb 14 2003

Keywords

• nonn

Comments

More generally let (x,y,z) be 3 positive integers and a(n) be the sequence a(1)=x, a(n) = a(n-1) + y if n is already in the sequence, a(n) = a(n-1) + z otherwise. Then it seems that a(n) is asymptotic to r*n where r is the largest positive root of q^2 = z*q + z - y.

Example: (x,y,z) = (2, 1, 2) gives A004956(n), (x,y,z) = (1, 2, 3) gives A007066(n). The present sequence is the case (1, 3, 2).

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,3,2)-hiccup sequence. - Michael De Vlieger, Jul 30 2025

Examples

			a(6)=13 hence a(13) = a(12) + 3 = 27 + 3 = 30.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 7.
• Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
• Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
• Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 2-3, 8.

Crossrefs

Cf. A004956, A007066, A026351, A079000. Apart from start, equals A080652 + 1.

Programs

Haskell

a064437 n = a064437_list !! (n-1)
a064437_list = 1 : f 2 [1] where
   f x zs@(z:_) = y : f (x + 1) (y : zs) where
     y = if x `elem` zs then z + 3 else z + 2
-- Reinhard Zumkeller, Sep 26 2014

Maple

A064437:= n -> ceil((1+sqrt(2))*(n-1)+1/(2+sqrt(2))):
seq(A064437(n),n=1..100); # Robert Israel, May 19 2014

Mathematica

a[1] = 1; a[n_] := a[n] = a[n-1] + If[MemberQ[Array[a, n-1], n], 3, 2];
Array[a, 100] (* Jean-François Alcover, Aug 01 2018 *)

PARI

an=vector(100); an[1]=1; a(n)=if(n<0,0,an[n]);
x=1; y=3; z=2; an[1]=x; for(n=2,100,an[n]=if(setsearch(Set(vector(n- 1,i,a(i))),n),a(n-1)+y,a(n-1)+z));
an

Formula

a(n) = ceiling((1+sqrt(2))*(n-1)+C) where C = 1/(2+sqrt(2)) = 0.292893218813...