A182760 Beatty sequence for (3 + 5^(-1/2))/2.

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117, 118, 120, 122, 124, 125, 127, 129, 130, 132, 134, 136, 137, 139, 141, 143, 144, 146, 148, 149, 151, 153, 155, 156, 158, 160, 162, 163, 165, 167, 168

Offset: 1

Author

Clark Kimberling, Nov 28 2010

Keywords

• nonn
• easy

Comments

Suppose that u and v are positive real numbers for which the sets S(u)={j*u} and S(v)={k*v}, for j>=1 and k>=1, are disjoint. Let a(n) be the position of n*u when the numbers in S(u) and S(v) are jointly ranked. Then, as is easy to prove, a is the Beatty sequence of the number r=1+u/v, and the complement of a is the Beatty sequence of s=1+v/u. For A182760, take u = golden ratio = (1+sqrt(5))/2 and v=sqrt(5), so that r=(3+5^(-1/2))/2 and s=(7-sqrt(5))/2.

Examples

			Let u=(1+sqrt(5))/2 and v=sqrt(5).  When the numbers ju and kv are jointly ranked, we write U for numbers of the form ju and V for the others.  Then the ordering of the ranked numbers is given by U V U V U U V U V U V U U ..  The positions of U are given by A182760.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A182761 (the complement of A182760), A242671

Programs

Magma

[Floor(n*(3+5^(-1/2))/2): n in [1..70]]; // Vincenzo Librandi, Oct 25 2011

Mathematica

Table[Floor[Sqrt[n/20]+3*n/2], {n,1,100}] (* G. C. Greubel, Jan 11 2018 *)

PARI

a(n)=floor(sqrt(n/20)+3*n/2) \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n) = floor(r*n), where r = (3 + 5^(-1/2))/2 = 1.72360...

Extensions

More than the usual number of terms are shown in order to distinguish this from a very similar sequence. - N. J. A. Sloane, Jan 20 2025