A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2. (Formerly M3299)

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

1,2

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.

Benoit Cloitre, The Golden Sieve and its connections to Hiccup sequences and Fraenkel games, arXiv:2602.17735 [math.NT], 2026. See pp. 4, 5, 27, 30.

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

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

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 *)

Prog

(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
(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

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).

Sequence in context: A189367 A310951 A310952 * A260395 A047537 A247985

Adjacent sequences: A007063 A007064 A007065 * A007067 A007068 A007069

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Mira Bernstein

Status

approved