A329975 Beatty sequence for 1 + x + x^2, where x is the real solution of 1/x + 1/(1+x+x^2) = 1.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 155, 159, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199, 203, 208, 212, 216, 220, 224, 228, 232

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, A329974 (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*(1+x+x^2)), where x = 1.324717... is the constant in A060006.