A004976 a(n) = floor(n*phi^3), where phi=(1+sqrt(5))/2.

0, 4, 8, 12, 16, 21, 25, 29, 33, 38, 42, 46, 50, 55, 59, 63, 67, 72, 76, 80, 84, 88, 93, 97, 101, 105, 110, 114, 118, 122, 127, 131, 135, 139, 144, 148, 152, 156, 160, 165, 169, 173, 177, 182, 186, 190, 194, 199

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is the position of the n-th 1 in the zero-one sequence [n*r+r]-[n*r]-[r], where r=sqrt(5); see A188221. Also, A004976=-1+A004958 (for n>=1), and A004976 is the complement of A188222. - Clark Kimberling, Mar 24 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
• Vincent Russo and Loren Schwiebert, Beatty Sequences, Fibonacci Numbers, and the Golden Ratio, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A000201, A001950, A004919, A004958, A188221, A188222.

Programs

Mathematica

r=5^(1/2); k=1;
t=Table[Floor[n*r+k*r]-Floor[n*r]-Floor[k*r], {n,1,220}]         (* A188221 *)
Flatten[Position[t,0] ]   (* A188222 *)
Flatten[Position[t,1] ]   (* A004976 *)
(* Clark Kimberling, Mar 24 2011 *)
With[{c=GoldenRatio^3},Floor[c*Range[0,50]]] (* Vincenzo Librandi, Apr 12 2012 *)

PARI

a(n)=2*n+sqrtint(5*n^2) \\ Charles R Greathouse IV, Apr 12 2012

Python

from math import isqrt
def A004976(n): return (isqrt(20*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 17 2022

Formula

a(n) = n+floor(2*n*phi). [Formula corrected by Clark Kimberling, Mar 22 2008]