A385177 a(n) = Sum_{k=1..n} ceiling(k/phi), where phi is the golden ratio (A001622).

1, 3, 5, 8, 12, 16, 21, 26, 32, 39, 46, 54, 63, 72, 82, 92, 103, 115, 127, 140, 153, 167, 182, 197, 213, 230, 247, 265, 283, 302, 322, 342, 363, 385, 407, 430, 453, 477, 502, 527, 553, 579, 606, 634, 662, 691, 721, 751, 782, 813, 845, 878, 911, 945, 979, 1014, 1050, 1086, 1123

Offset: 1

Author

Paolo Xausa, Jun 20 2025

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001622, A183136.

Partial sums of A019446.

Programs

Mathematica

Accumulate[Ceiling[Range[100]/GoldenRatio]]

Python

from math import isqrt
def A385177(n): return n+sum(isqrt(5*i**2)-i>>1 for i in range(1,n+1)) # Chai Wah Wu, Jun 20 2025

Formula

a(n) = A183136(n) + n.