A004956 a(n) = ceiling(n*phi), where phi is the golden ratio, A001622.

0, 2, 4, 5, 7, 9, 10, 12, 13, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 89

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(0)=0, a(1)=2; for n > 1, a(n) = a(n-1) + 2 if n is already in the sequence, a(n) = a(n-1) + 1 otherwise.

Integer solutions to the equation x = ceiling(phi*floor(x/phi)). - Benoit Cloitre, Feb 14 2004

From Benoit Cloitre, Mar 05 2007: (Start)

The following is an alternative way to obtain this sequence. NP means "term not in parentheses". Write down the natural numbers and mark the least NP, which is 1:

1* 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...

Take the first NP (which is 1) and parenthesize it; mark the least NP (which is 2):

(1*) 2* 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...

Take the 2nd NP (which is 3) and parenthesize it; mark the next NP (which is 4):

(1*) 2* (3) 4* 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...

Take the 4th NP (which is 6) and parenthesize it; mark the next NP (which is 5):

(1*) 2* (3) 4* 5* (6) 7 8 9 10 11 12 13 14 15 16 17 18 19 ...

Continuing in this way we obtain

(1*) 2* (3) 4* 5* (6) 7* (8) 9* 10* (11) 12* 13* (14) 15* (16) 17* (18) 19* ...

The starred entries (after the first) give the sequence. (End)

From Rick L. Shepherd, Dec 05 2009: (Start)

An equivalent statement of the sieving process described by Benoit Cloitre on Mar 05 2007:

Begin with the natural numbers N. Repeatedly perform these two steps:

i) Let k = N's least remaining term not yet used in Step ii).

ii) Remove the k-th remaining term from N.

The remaining terms of N are the (positive) terms shared by this sequence and A026351.

The terms removed from N (the complement) are A026352's terms (see also A004957).

The PARI program performs this sieving process and prints the positive terms of this sequence. (End)

In the Fokkink-Joshi paper, apart from a(0) = 0, this sequence is the Cloitre (0,2,2,1)-hiccup sequence, i.e., a(1) = 2; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+1. - Michael De Vlieger, Jul 30 2025

Links

• Christian Krause, Table of n, a(n) for n = 0..10000
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.
• 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 p. 3.

Crossrefs

Cf. A001622, A004957, A026352 (Cloitre (1,1,2,3)-hiccup sequence).

Essentially same as A026351.

Programs

Mathematica

Ceiling[Range[0,100]GoldenRatio] (* Paolo Xausa, Oct 28 2023 *)
(* Second program: *)
cloitreH[j_, x_, y_, z_, w_ : 120] := Module[{c, k}, c[] := False; k = x; c[x] = True; {x}~Join~Reap[Do[If[c[n - j], k += y, k += z]; c[k] = True; Sow[k], {n, 2, w}] ][[-1, 1]] ]; {0}~Join~cloitreH[0, 2, 2, 1] (* _Michael De Vlieger, Jul 30 2025 *)

PARI

{/* paws = Print Absolute values of all elements in vector v With same Sign as sn */
paws(v,sn) = for(m=1,matsize(v)[2], if(sign(v[m])==sign(sn),\
print1(abs(v[m]),",")))}
{/* Sieve through lim numbers; make values negative to signify "removed" */
lim=100; v=vector(lim,i,i); i=0; j=0; c=1;
while(i0, k=v[i]; c=c--;
while(c0, c++)); v[j]=-v[j])); paws(v,1)\
/* Changing "1" to "-1" in paws() above prints out the terms of A026352 instead */} \\ Rick L. Shepherd, Dec 05 2009

PARI

a(n) = ceil(n*(1 + sqrt(5))/2); \\ Michel Marcus, Apr 13 2021