|
%I M0622 #64 Feb 14 2024 17:31:24
%S 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,
%T 40,42,44,45,47,49,50,52,53,55,57,58,60,61,63,65,66,68,70,71,73,74,76,
%U 78,79,81,83,84,86,87,89,91,92,94,95,97,99,100,102,104,105
%N Nearest integer to n*tau where tau = (1+sqrt(5))/2.
%C First column of inverse Stolarsky array.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Iain Fox, <a href="/A007067/b007067.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Vincenzo Librandi)
%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
%H Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%F Satisfies a(a(n)) = a(n) + n. - _Franklin T. Adams-Watters_, Aug 14 2006
%F a(n) = floor((A066096(2*n) + 1)/2). - _Peter Munn_, Jan 12 2018
%p A007067:=n->round(n*(1+sqrt(5))/2); seq(A007067(n), n=0..100); # _Wesley Ivan Hurt_, Nov 27 2013
%t a[n_] := Round[n*GoldenRatio]; Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Jun 27 2012 *)
%o (PARI) a(n) = round(n*(1+sqrt(5))/2) \\ _Michel Marcus_, May 20 2013
%o (Python)
%o from math import isqrt
%o def A007067(n): return (isqrt(5*n**2<<2)>>1)+n+1>>1 # _Chai Wah Wu_, Aug 26 2022
%Y Cf. A166946 (characteristic function), A007064 (complement).
%Y Different from A026355.
%Y Sequences with similar terms: A022342, A295282.
%Y Other roundings of n*tau: A000201, A004956, A066096.
%Y Cf. A000045 (Fibonacci numbers), A001622 (value of tau).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Mira Bernstein_
|
