A249115 Floor(r*n), where r = (5 - sqrt(5))/2; the Beatty complement of A003231.

1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 89, 91

Offset: 1

Author

Clark Kimberling, Oct 21 2014

Keywords

• nonn
• easy

Comments

Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A249115(n) is the position of n*(tau - 1) in the ordered union of R and S.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.

Crossrefs

Cf. A003231, A001622.

Programs

Magma

[Floor(n*(5-Sqrt(5))/2): n in [1..100]]; // Vincenzo Librandi, Oct 25 2014

Mathematica

Table[Floor[(5 - Sqrt[5])/2*n], {n, 1, 200}]