A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.

1, 4, 7, 9, 12, 15, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 70, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 98, 101, 104, 106, 109, 111, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145, 148, 151, 153, 156, 159, 161, 164, 166

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

First column of dual Wythoff array, A126714.

Positions of 0's in A189479.

Skala (2016) asks if this sequence also gives the positions of the 0's in A283310. - N. J. A. Sloane, Mar 06 2017

Upper Wythoff sequence plus 2, when shifted by 1. - Michel Dekking, Aug 26 2019

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,2,3)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+3. - Michael De Vlieger, Jul 30 2025

References

• Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
• D. R. Morrison, "A Stolarsky array of Wythoff pairs," in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
• 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
• Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 9.
• Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 3-4, 7, 10.
• Clark Kimberling, Interspersions
• Matthew Skala, Graph Nimors, arXiv preprint arXiv:1604.04072 [math.CO], 2016.
• N. J. A. Sloane, Classic Sequences

Crossrefs

Cf. A064437.

Apart from initial terms, same as A026356 (Cloitre (0,2,2,3)-hiccup sequence).

First column of A126714.

Complement is (essentially) A026355.

Equals 1 + A004957, also n + A004956.

First differences give A076662.

Complement of A099267. [Gerald Hillier, Dec 19 2008]

Cf. A193214 (primes). Except for the first term equal to A001950 + 2.

Cf. A026352 (Cloitre (1,1,2,3)-hiccup sequence), A064437 (Cloitre (0,1,3,2)-hiccup sequence).

Programs

Haskell

a007066 n = a007066_list !! (n-1)
a007066_list = 1 : f 2 [1] where
   f x zs@(z:_) = y : f (x + 1) (y : zs) where
     y = if x `elem` zs then z + 2 else z + 3
-- Reinhard Zumkeller, Sep 26 2014, Sep 18 2011

Maple

Digits := 100: t := (1+sqrt(5))/2; A007066 := proc(n) if n <= 1 then 1 else floor(1+t*floor(t*(n-1)+1)); fi; end;

Mathematica

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)
With[{grs=GoldenRatio^2},Table[1+Ceiling[grs(n-1)],{n,70}]] (* Harvey P. Dale, Jun 24 2011 *)

Python

from math import isqrt
def A007066(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n if n > 1 else 1 # Chai Wah Wu, Aug 25 2022

Formula

a(n) = floor(1+phi*floor(phi*(n-1)+1)), phi=(1+sqrt(5))/2, n >= 2.

a(1)=1; for n>1, a(n)=a(n-1)+2 if n is already in the sequence, a(n)=a(n-1)+3 otherwise. - Benoit Cloitre, Mar 06 2003

a(n+1) = floor(n*phi^2) + 2, n>=1. - Michel Dekking, Aug 26 2019