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

0,2

Comments

For n>=1, a(n) is the position of the n-th 1 in the zero-one sequence [nr+r]-[nr]-[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

Formula

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A311254 A311255 A190885 * A341254 A311256 A311257

Adjacent sequences: A004973 A004974 A004975 * A004977 A004978 A004979

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved