A330118 - OEIS

A330118

Beatty sequence for 1+x+x^2, where 1/(1+x) + 1/(1+x+x^2) = 1.

2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 69, 72, 74, 76, 79, 81, 83, 86, 88, 90, 92, 95, 97, 99, 102, 104, 106, 109, 111, 113, 116, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Let x be the positive solution of 1/(1+x) + 1/(1+x+x^2) = 1. Then (floor(n*(1+x)) and (floor(n*(1+x+x^2))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

LINKS

Table of n, a(n) for n=1..61.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

FORMULA

a(n) = floor(n (1+x))), where x = 0.7548776662... is the constant in A075778.

MATHEMATICA

r = x /. FindRoot[1/(1 + x) + 1/(1 + x + x^2) == 1, {x, 1, 2}, WorkingPrecision -> 200]

RealDigits[r] (* A075778 *)

Table[Floor[n*(1 + r)], {n, 1, 250}] (* A330117 *)

Table[Floor[n*(1 + r + r^2)], {n, 1, 250}] (* A330118 *)

Plot[1/(1 + x) + 1/(1 + x + x^2) - 1, {x, 0, 2}]