A329974 Beatty sequence for the real solution x of 1/x + 1/(1+x+x^2) = 1.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87

Offset: 1

Author

Clark Kimberling, Jan 02 2020

Keywords

• nonn
• easy

Comments

Let x be the real solution of 1/x + 1/(1+x+x^2) = 1. Then (floor(n*x)) and (floor(n*(x^2 + x + 1))) 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

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

Crossrefs

Cf. A329825, A060006, A329975 (complement).

Programs

Mathematica

Solve[1/x + 1/(1 + x + x^2) == 1, x]
u = 1/3 (27/2 - (3 Sqrt[69])/2)^(1/3) + (1/2 (9 + Sqrt[69]))^(1/3)/3^(2/3);
u1 = N[u, 150]
RealDigits[u1, 10][[1]]  (* A060006 *)
Table[Floor[n*u], {n, 1, 50}]              (* A329974 *)
Table[Floor[n*(1 + u + u^2)], {n, 1, 50}]  (* A329975 *)
Plot[1/x + 1/(1 + x + x^2) - 1, {x, -2, 2}]

Formula

a(n) = floor(n*x), where x = 1.324717... is the constant in A060006.