A330117 -id:A330117 - cp's OEIS frontend

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

Offset: 1

Clark Kimberling, Jan 04 2020

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.

Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A329825, A075778, A330117 (complement).

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}]

a(n) = floor(n*(1+x+x^2)), where x = 0.7548776662... is the constant in A075778.

cp's OEIS Frontend

A330118 Beatty sequence for 1+x+x^2, where 1/(1+x) + 1/(1+x+x^2) = 1.

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