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

0,3

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

Formula

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

Mathematica

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

Prog

(PARI) a(n) = (n+sqrtint(5*n^2))\2;
[a(n)|n<-[0..100]] \\ Simon Strandgaard, Jun 28 2022
(Magma) [Floor((1+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023
(SageMath) [floor(golden_ratio*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

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.

Sequence in context: A285676 A085270 A330063 * A000201 A090908 A292644

Adjacent sequences: A066093 A066094 A066095 * A066097 A066098 A066099

Keyword

nonn,easy

Author

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

Extensions

Name corrected by Peter Munn, Dec 06 2017

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

Status

approved