A330002 Beatty sequence for x, where 1/x + 1/(x+1)^2 = 1.

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83

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.

Differs from A047201 first at A047201(85)=106, a(85)=105. - R. J. Mathar, Jan 11 2020

Links

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

Crossrefs

Cf. A329825, A255249, A330003 (complement).

Programs

Maple

A330002 := proc(n)
    local x;
    x := -2*cos(5*Pi/7) ;
    floor(n*x) ;
end proc: # R. J. Mathar, Jan 11 2020

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.