A330003 Beatty sequence for (x+1)^2, where 1/x + 1/(x+1)^2 = 1.

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 106, 111, 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, 176, 181, 186, 191, 196, 201, 207, 212, 217, 222, 227, 232, 237, 242, 247, 252, 257, 262, 267, 272, 277, 282

Offset: 1

Author

Clark Kimberling, Jan 04 2020

Keywords

• nonn
• easy

Comments

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

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

Crossrefs

Cf. A329825, A255249, A330002 (complement).

Programs

Mathematica

r = x /. FindRoot[1/x + 1/(x+1)^2 == 1, {x, 2, 10}, WorkingPrecision -> 120]
RealDigits[r][[1]] (* A255249 *)
Table[Floor[n*r], {n, 1, 250}]        (* A330002 *)
Table[Floor[n*(1+r)^2], {n, 1, 250}]  (* A330003 *)

Formula

a(n) = floor(n x), where x = 1.24697960371... is the constant in A255249.