A026356 a(n) = floor((n-1)*phi) + n + 1, n > 0, where phi = (1+sqrt(5))/2.

2, 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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Greatest k such that s(k) = n+1, where s = A026354.

Positions of 1 in A189661.

a(n+1) = A001950(n)-2, the Upper Wythoff sequence shifted by 2. - Michel Dekking, Oct 18 2018

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,2,2,3)-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)+3. - Michael De Vlieger, Jul 28 2025

Links

• Michel Dekking, Table of n, a(n) for n = 1..1000
• 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. 7, 10, 17.

Crossrefs

Cf. A000201, A026351, etc. Apart from initial terms, same as A007066. Complement is A189662, closely related to A026355.

Programs

Mathematica

(* See A189661. Second program: *)
cloitreH[j_, x_, y_, z_, w_ : 120] := Block[{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]] ]; cloitreH[0, 2, 2, 3] (* _Michael De Vlieger, Jul 28 2025 *)

PARI

r = (1 + sqrt(5))/2;
a(n) = if(n<1, 1, floor((n - 1)* r) + n + 1);
for(n=1, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 25 2017

Python

from sympy import sqrt
import math
r=(1 + sqrt(5))/2
def a(n): return 1 if n<1 else int(math.floor((n - 1)*r)) + n + 1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

Python

from math import isqrt
def A026356(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n # Chai Wah Wu, Aug 11 2022

Extensions

Data corrected by Michel Dekking, Oct 18 2018