A329847 Beatty sequence for (3+sqrt(89))/8.

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 101, 102

Offset: 1

Author

Clark Kimberling, Jan 02 2020

Keywords

• nonn
• easy

Comments

Let r = (3+sqrt(89))/8. Then (floor(n*r)) and (floor(n*r + 5r/4)) 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, A329848 (complement).

Programs

Mathematica

t = 5/4; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329847 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329848 *)

Formula

a(n) = floor(n*r), where r = (3+sqrt(89))/8.