A004919 a(n) = floor(n*phi^4), where phi is the golden ratio, A001622.

0, 6, 13, 20, 27, 34, 41, 47, 54, 61, 68, 75, 82, 89, 95, 102, 109, 116, 123, 130, 137, 143, 150, 157, 164, 171, 178, 185, 191, 198, 205, 212, 219, 226, 233, 239, 246, 253, 260, 267, 274, 281, 287, 294, 301, 308

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The golden section or golden ratio is now usually denoted by "phi", but it in the older literature it was more often denoted by "tau." - N. J. A. Sloane, Feb 17 2013

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.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A004920, A004921, A004922, A004923, A004924, A004925, A004926.

Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.

Cf. A004934, A004935, A004976, A066096, A090909.

Cf. A000201, A001950.

Programs

Magma

[Floor((7+3*Sqrt(5))*n/2): n in [0..60]]; // G. C. Greubel, Aug 22 2023

Mathematica

With[{c=GoldenRatio^4},Floor[c*Range[0,50]]] (* Harvey P. Dale, Apr 11 2012 *)

Python

from math import isqrt
def A004919(n): return (3*n+isqrt(45*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 17 2022

SageMath

[floor(golden_ratio^4*n) for n in range(61)] # G. C. Greubel, Aug 22 2023

Formula

a(n) = 2*n + floor(3*n*tau). [Formula corrected by Charles R Greathouse IV, Mar 11 2011]