A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

0, 2, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy

Comments

First column of inverse Stolarsky array.

A rectangle of size a(n) X n approximates a golden rectangle. So does A295282(n) X n, which targets the golden ratio's underlying objective. These approximations differ first for n = 4 and generally if n = F(6*k)/2, where F(n) = A000045(n) is the n-th Fibonacci number and k >= 1. - Peter Munn, Jan 12 2018

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Iain Fox, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vincenzo Librandi)
• Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
• Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• N. J. A. Sloane, Classic Sequences.

Crossrefs

Cf. A166946 (characteristic function), A007064 (complement).

Different from A026355.

Sequences with similar terms: A022342, A295282.

Other roundings of n*tau: A000201, A004956, A066096.

Cf. A000045 (Fibonacci numbers), A001622 (value of tau).

Programs

Maple

A007067:=n->round(n*(1+sqrt(5))/2); seq(A007067(n), n=0..100); # Wesley Ivan Hurt, Nov 27 2013

Mathematica

a[n_] := Round[n*GoldenRatio]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 27 2012 *)

PARI

a(n) = round(n*(1+sqrt(5))/2) \\ Michel Marcus, May 20 2013

Python

from math import isqrt
def A007067(n): return (isqrt(5*n**2<<2)>>1)+n+1>>1 # Chai Wah Wu, Aug 26 2022

Formula

Satisfies a(a(n)) = a(n) + n. - Franklin T. Adams-Watters, Aug 14 2006

a(n) = floor((A066096(2*n) + 1)/2). - Peter Munn, Jan 12 2018