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A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.

%I A293321 #6 Oct 09 2017 08:25:03
%S A293321 3,5,10,21,42,84,168,335,670,1340,2681,5362,10723,21447,42894,85788,
%T A293321 171575,343151,686302,1372604,2745208,5490415,10980830,21961661,
%U A293321 43923322,87846643,175693287,351386574,702773148,1405546295,2811092590,5622185181,11244370361
%N A293321 The integer k that minimizes |k/2^n - tau^2|, where tau = (1+sqrt(5))/2 = golden ratio.
%H A293321 Clark Kimberling, <a href="/A293321/b293321.txt">Table of n, a(n) for n = 0..1000</a>
%F A293321 a(n) = floor(1/2 + r*2^n), where r = (3+sqrt(5))/2.
%F A293321 a(n) = A293319(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293316(n).
%t A293321 z = 120; r = 1+GoldenRatio;
%t A293321 Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
%t A293321 Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
%t A293321 Table[Round[r*2^n], {n, 0, z}]; (* A293321 *)
%o A293321 (PARI) a(n) = (2^n*(3+sqrt(5))+1)\2; \\ _Altug Alkan_, Oct 08 2017
%Y A293321 Cf. A001622, A293313, A293319, A293320.
%K A293321 nonn,easy
%O A293321 0,1
%A A293321 _Clark Kimberling_, Oct 07 2017