A330117 - OEIS

A330117

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

1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 49, 50, 52, 54, 56, 57, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 107, 108, 110, 112

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

OFFSET

1,2

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..64.

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}]