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A293405 The integer k that minimizes |k/n^2 - tau^2|, where tau = (1+sqrt(5))/2 (golden ratio).

%I A293405 #4 Oct 11 2017 21:57:05
%S A293405 0,3,10,24,42,65,94,128,168,212,262,317,377,442,513,589,670,757,848,
%T A293405 945,1047,1155,1267,1385,1508,1636,1770,1909,2053,2202,2356,2516,2681,
%U A293405 2851,3026,3207,3393,3584,3780,3982,4189,4401,4618,4841,5069,5302,5540,5783
%N A293405 The integer k that minimizes |k/n^2 - tau^2|, where tau = (1+sqrt(5))/2 (golden ratio).
%H A293405 Clark Kimberling, <a href="/A293405/b293405.txt">Table of n, a(n) for n = 0..1000</a>
%F A293405 a(n) = floor(1/2 + tau*n^2).
%F A293405 a(n) = A293403(n) if (fractional part of (1+tau)*n^2) < 1/2, else a(n) = A293404(n).
%t A293405 z = 120; r = 1+GoldenRatio;
%t A293405 Table[Floor[r*n^2], {n, 0, z}];   (* A293403 *)
%t A293405 Table[Ceiling[r*n^2], {n, 0, z}]; (* A293404 *)
%t A293405 Table[Round[r*n^2], {n, 0, z}];   (* A293405 *)
%Y A293405 Cf. A001622, A293402, A293403, A293404.
%K A293405 nonn,easy
%O A293405 0,2
%A A293405 _Clark Kimberling_, Oct 11 2017