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Greatest integer k such that k/n^2 < (1 + sqrt(5))/2 (the golden ratio).
3

%I #12 Sep 08 2022 08:46:19

%S 0,1,6,14,25,40,58,79,103,131,161,195,232,273,317,364,414,467,524,584,

%T 647,713,783,855,931,1011,1093,1179,1268,1360,1456,1554,1656,1762,

%U 1870,1982,2096,2215,2336,2461,2588

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

%H Clark Kimberling, <a href="/A293400/b293400.txt">Table of n, a(n) for n = 0..1000</a>

%H Felipe Gonçalves, Diogo Oliveira e Silva, João P. G. Ramos, <a href="https://arxiv.org/abs/2003.10771">New Sign Uncertainty Principles</a>, arXiv:2003.10771 [math.CA], 2020.

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

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

%t z = 120; r = GoldenRatio;

%t Table[Floor[r*n^2], {n, 0, z}]; (* A293400 *)

%t Table[Ceiling[r*n^2], {n, 0, z}]; (* A293401 *)

%t Table[Round[r*n^2], {n, 0, z}]; (* A293402 *)

%o (Magma) [Floor((1 + Sqrt(5))/2*n^2) : n in [0..80]]; // _Wesley Ivan Hurt_, Jul 03 2020

%Y Cf. A001622, A293401, A293402.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Oct 11 2017