A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2. (Formerly M0622)

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

0,2

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.

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

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 *)

Prog

(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

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).

Sequence in context: A029921 A026355 A099267 * A186322 A092979 A135260

Adjacent sequences: A007064 A007065 A007066 * A007068 A007069 A007070

Keyword

nonn,easy

Author

N. J. A. Sloane, Mira Bernstein

Status

approved