A376448 a(n) = k if k is odd otherwise a(n) = k+1 and k = floor( 2^n*(1+sqrt(5))/2 ).

1, 3, 7, 13, 25, 51, 103, 207, 415, 829, 1657, 3313, 6627, 13255, 26509, 53019, 106039, 212079, 424157, 848315, 1696631, 3393263, 6786527, 13573053, 27146105, 54292211, 108584423, 217168845, 434337691, 868675383, 1737350767, 3474701533, 6949403065, 13898806131, 27797612261, 55595224523

Offset

0,2

Comments

The sequence of all multiples of an irrational b is equidistributed modulo 1. Such a sequence is called a Weyl sequence. It is common practice in computing to approximate a Weyl sequence by taking integer multiples of some integer m modulo a power of two. This requires that the integer m is odd. This sequence provides suitable m = a(n) for the case modulo 2^n. It utilizes the golden ratio for approximation of irrationality.

Links

Table of n, a(n) for n=0..35.

Formula

a(n) = 2*A293313(n-1) + 1, for n > 0.

Example

An example for a pseudo Weyl sequence obtained from a(3):
{0, 1, 2, 3, 4, 5, 6, 7} * a(3) mod 2^3 = {0, 5, 2, 7, 4, 1, 6, 3}. (Without zero also part of A194868).

Mathematica

k[n_]:=Floor[2^n*GoldenRatio]; Table[If[OddQ[k[n]], k[n], k[n]+1], {n, 0, 35}] (* James C. McMahon, Oct 20 2024 *)

Prog

(PARI) a(n) = {my( m=floor(quadgen(5)<<n) ); if(m%2==0, return(m+1), return(m))}

Crossrefs

Cf. A054065, A194868, A293313, A293314, A293315.

Sequence in context: A078000 A190569 A007909 * A363143 A282913 A284026

Adjacent sequences: A376445 A376446 A376447 * A376449 A376450 A376451

Keyword

nonn,easy

Author

Thomas Scheuerle, Sep 23 2024

Status

approved