A293400 Greatest integer k such that k/n^2 < (1 + sqrt(5))/2 (the golden ratio).

0, 1, 6, 14, 25, 40, 58, 79, 103, 131, 161, 195, 232, 273, 317, 364, 414, 467, 524, 584, 647, 713, 783, 855, 931, 1011, 1093, 1179, 1268, 1360, 1456, 1554, 1656, 1762, 1870, 1982, 2096, 2215, 2336, 2461, 2588, 2719, 2854, 2991, 3132, 3276, 3423, 3574, 3727, 3884

Offset

0,3

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Felipe Gonçalves, Diogo Oliveira e Silva, and João P. G. Ramos, New Sign Uncertainty Principles, arXiv:2003.10771 [math.CA], 2020-2023.

Formula

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

a(n) = A293401(n) - 1 for n > 0.

Mathematica

z = 120; r = GoldenRatio;
Table[Floor[r*n^2], {n, 0, z}];   (* A293400 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293401 *)
Table[Round[r*n^2], {n, 0, z}];   (* A293402 *)

Prog

(Magma) [Floor((1 + Sqrt(5))/2*n^2) : n in [0..80]]; // Wesley Ivan Hurt, Jul 03 2020
(Python)
from math import isqrt
def A293400(n): return ((n**2+isqrt(5*n**4))>>1) # Aidan Chen, Jan 10 2026

Crossrefs

Cf. A001622, A293401, A293402.

Sequence in context: A185594 A095794 A119867 * A026055 A227952 A288939

Adjacent sequences: A293397 A293398 A293399 * A293401 A293402 A293403

Keyword

nonn,easy

Author

Clark Kimberling, Oct 11 2017

Status

approved