login
Greatest integer k such that k/2^n < tau^2, where tau = (1+sqrt(5))/2 = golden ratio.
3

%I #8 Sep 08 2022 08:46:19

%S 2,5,10,20,41,83,167,335,670,1340,2680,5361,10723,21446,42893,85787,

%T 171575,343150,686301,1372603,2745207,5490415,10980830,21961660,

%U 43923321,87846643,175693286,351386573,702773147,1405546295,2811092590,5622185180,11244370361

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

%H Clark Kimberling, <a href="/A293319/b293319.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = A293320(n) - 1.

%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 (Magma) [Floor((2^n*(3+Sqrt(5)))/2): n in [0..33]]; // _Vincenzo Librandi_, Oct 08 2017

%Y Cf. A001622, A293313, A293320, A293321.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Oct 07 2017