%I #6 Oct 09 2017 08:25:03

%S 3,5,10,21,42,84,168,335,670,1340,2681,5362,10723,21447,42894,85788,

%T 171575,343151,686302,1372604,2745208,5490415,10980830,21961661,

%U 43923322,87846643,175693287,351386574,702773148,1405546295,2811092590,5622185181,11244370361

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

%H Clark Kimberling, <a href="/A293321/b293321.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = floor(1/2 + r*2^n), where r = (3+sqrt(5))/2.

%F a(n) = A293319(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293316(n).

%t z = 120; r = 1+GoldenRatio;

%t Table[Floor[r*2^n], {n, 0, z}]; (* A293319 *)

%t Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)

%t Table[Round[r*2^n], {n, 0, z}]; (* A293321 *)

%o (PARI) a(n) = (2^n*(3+sqrt(5))+1)\2; \\ _Altug Alkan_, Oct 08 2017

%Y Cf. A001622, A293313, A293319, A293320.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Oct 07 2017