A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.

0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106

Offset: 0

Author

Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001

Keywords

• nonn
• easy

Comments

a(n) is the smallest number different from a(i) and a(i)+i for i < n.

The losing positions in the game of Wythoff-Nim are precisely the pairs (a(n), a(n)+n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Essentially the partial sums of A001468.

Cf. A000201, A001622, A003622, A022342, A026351, A035336, A060143.

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.

Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.

Cf. A004933, A004934, A004935, A004976, A090909.

Programs

Magma

[Floor((1+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

Mathematica

Floor[GoldenRatio*Range[0, 80]] (* G. C. Greubel, Sep 12 2023 *)

PARI

a(n) = (n+sqrtint(5*n^2))\2;
[a(n)|n<-[0..100]] \\ Simon Strandgaard, Jun 28 2022

SageMath

[floor(golden_ratio*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

Formula

For n >= 1, a(n) = A000201(n).

Duplicate values in A060143.

a(n) = 1 + A022342(n) = A000201(n).

a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2. - Peter Munn, Jan 12 2018

a(n) = A026351(n) - 1. - Philippe Deléham, Jan 15 2023

Extensions

Name corrected by Peter Munn, Dec 06 2017

New name using a formula from Peter Munn by Peter Luschny, Jan 18 2023