A026352 a(n) = floor(n*tau) + n + 1 where tau is the golden ratio A001622.

1, 3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, 35, 37, 40, 42, 45, 48, 50, 53, 55, 58, 61, 63, 66, 69, 71, 74, 76, 79, 82, 84, 87, 90, 92, 95, 97, 100, 103, 105, 108, 110, 113, 116, 118, 121, 124, 126, 129, 131, 134, 137, 139, 142, 144

Offset: 0

Author

Clark Kimberling, Dec 11 1999

Keywords

• nonn

Comments

a(n) = greatest k such that s(k) = n+1, where s = A026350.

Indices at which blocks (0;1) occur in infinite Fibonacci word; i.e., n such that A005614(n)=0 and A005614(n+1)=1. - Benoit Cloitre, Nov 15 2003

Except for the first term, these are the numbers whose lazy Fibonacci representation (see A095791) includes both 1 and 2; thus this sequence is a subsequence of the lower Wythoff sequence, A000201. - Clark Kimberling, Jun 10 2004 [A-number typo corrected by Nathan Fox, May 03 2014]

a(n) = n-th number k whose lazy Fibonacci representation (as in A095791) has more summands than that of k-1. - Clark Kimberling, Jun 12 2004

a(n) = position of n-th 0 in A096270. - Clark Kimberling, Apr 22 2011

Maximum number of chips in a pile created at each step in the game described by Roland Schroeder in his comment at A000201. (From Allan C. Wechsler via Seqfan.)

In the Fokkink-Joshi paper, this sequence is the Cloitre (1,1,2,3)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n-1, else a(n) = a(n-1)+3. - Michael De Vlieger, Jul 28 2025

Links

• 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. 8, 17.
• Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.
• Urban Larsson and Nathan Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.
• Ali Sada, Should A000201 and A026352 be cross-referenced?, Seqfan thread, Jun 2023.

Crossrefs

Essentially same as A004957.

Cf. A001622, A005614, A026350, A095791, A096270.

Subsequence of A000201.

Complement of A026351.

Programs

Mathematica

Table[Floor[GoldenRatio*n]+n+1,{n,0,60}] (* Harvey P. Dale, Aug 24 2021 *)
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[1, 1, 2, 3] (* _Michael De Vlieger, Jul 28 2025 *)

PARI

a(n) = floor(n*(sqrt(5)+1)/2) + n + 1; \\ Michel Marcus, Sep 15 2016

Python

from math import isqrt
def A026352(n): return (n+isqrt(5*n**2)>>1)+n+1 # Chai Wah Wu, Aug 25 2022

Formula

a(n) = A026351(n)+n. - R. J. Mathar, Jun 24 2025