A183136 a(n) = [1/r]+[2/r]+...+[n/r], where r = golden ratio = (1+sqrt(5))/2 and []=floor.

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 50, 58, 67, 76, 86, 97, 108, 120, 132, 145, 159, 173, 188, 204, 220, 237, 254, 272, 291, 310, 330, 351, 372, 394, 416, 439, 463, 487, 512, 537, 563, 590, 617, 645, 674, 703, 733, 763, 794, 826, 858

Offset: 1

Author

Clark Kimberling, Dec 26 2010

Keywords

• nonn

Comments

a(n) + A183137(n) = A000217(n) (the triangular numbers).

Examples

			The terms [k/r] are given by A060143 (and A005206): 0,1,1,2,3,3,4,4,5,6,6,7,8,8,...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Hector Zenil, N. Kiani, and J. Tegner, Low Algorithmic Complexity Entropy-deceiving Graphs, arXiv preprint arXiv:1608.05972 [cs.IT], 2016-2017.

Crossrefs

Cf. A183137, A000217, A005206, A060143, A060144.

Programs

Mathematica

Accumulate[Floor[Range[100]/GoldenRatio]] (* Paolo Xausa, Jun 20 2025 *)

PARI

default(realprecision,100); r=(1+sqrt(5))/2; for(n=1,99, print1(sum(k=1,n,floor(k/r)), ", "))

Python

from math import isqrt
def A183136(n): return sum(isqrt(5*i**2)-i>>1 for i in range(1,n+1)) # Chai Wah Wu, Jun 20 2025

Formula

a(n) = [1/r]+[2/r]+...+[n/r], where r = golden ratio = (1+sqrt(5))/2 and []=floor.