A007064 Numbers not of form "nearest integer to n*tau", tau = (1+sqrt(5))/2.

1, 4, 7, 9, 12, 14, 17, 20, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 48, 51, 54, 56, 59, 62, 64, 67, 69, 72, 75, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 103, 106, 109, 111, 114, 117, 119, 122, 124, 127, 130, 132, 135

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

First column of Stolarsky array.

This sequence and A057843 are very similar - this can be seen if the terms equal to 4 are aligned. - Thomas Baruchel, Nov 04 2003

References

• Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv:2312.11706 [math.CO], 2023.
• Clark Kimberling, Interspersions
• Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society 117 (1993) 313-321.
• N. J. A. Sloane, Classic Sequences
• K. B. Stolarsky, A set of generalized Fibonacci sequences such that each natural number belongs to exactly one, Fib. Quart., 15 (1977), 224.

Crossrefs

Complement of A007067.

Cf. A001622, A035506.

Programs

Mathematica

max = 100; Complement[ Range[ max*GoldenRatio], Round[ Range[max]*GoldenRatio]] (* Jean-François Alcover, Oct 10 2011 *)

PARI

a(n) = tau=(1+sqrt(5))/2; floor(n*(1+tau) - tau/2) \\ Michel Marcus, May 21 2013

Python

from math import isqrt
def A007064(n): return (6*n-1+isqrt(5*((n<<1)-1)**2))>>2 # Chai Wah Wu, Feb 11 2025

Formula

a(n) = floor[n*(1+tau)-tau/2] =floor[n*2.6180...-0.8090...]. - Henry Bottomley, Sep 03 2001