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

3, 5, 10, 21, 42, 84, 168, 335, 670, 1340, 2681, 5362, 10723, 21447, 42894, 85788, 171575, 343151, 686302, 1372604, 2745208, 5490415, 10980830, 21961661, 43923322, 87846643, 175693287, 351386574, 702773148, 1405546295, 2811092590, 5622185181, 11244370361

Offset: 0

Author

Clark Kimberling, Oct 07 2017

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001622, A293313, A293319, A293320.

Programs

Mathematica

z = 120; r = 1+GoldenRatio;
Table[Floor[r*2^n], {n, 0, z}];   (* A293319 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293320 *)
Table[Round[r*2^n], {n, 0, z}]; (* A293321 *)

PARI

a(n) = (2^n*(3+sqrt(5))+1)\2; \\ Altug Alkan, Oct 08 2017

Formula

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

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