A352676 Intersection of Beatty sequences for sqrt(3) and 1+sqrt(3).

5, 8, 10, 13, 19, 24, 27, 32, 38, 43, 46, 51, 57, 60, 62, 65, 71, 76, 79, 81, 84, 90, 95, 98, 103, 109, 112, 114, 117, 122, 128, 131, 133, 136, 142, 147, 150, 152, 155, 161, 166, 169, 174, 180, 183, 185, 188, 193, 199, 202, 204, 207, 213, 218, 221, 226, 232

Offset: 1

Author

Clark Kimberling, Mar 26 2022

Keywords

• nonn

Comments

Conjectures:

(1) a(n+1)-a(n) is in {2,3,4,5,6} for every n, and each of these differences occurs infinitely many times.

(2) Limit_{n->oo} a(n)/n = (3/2)*(1+sqrt(3)).

(3) Let d(n) = a(n) - A352673(n); then d(n) = 0 for infinitely many n, but {d(n)} is unbounded below and above.

Examples

			The two Beatty sequences, (1,3,5,6,8,10,12,13,15,17,19,20,...) and (2,5,8,10,13,16,19,21,24,...), share the numbers (5,8,10,13,19,24,...).
		

Links

• Clark Kimberling, R. Stanley, A. Kalmynin, Limit associated with two Beatty sequences that are not a Beatty pair, Math Overflow, Mar 2022.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A022838, A054088.

Programs

Mathematica

z = 200; r = Sqrt[3]; s = 1 + Sqrt[3];
u = Table[Floor[n r], {n, 1, z}]    (* A022838 *)
v = Table[Floor[n s], {n, 1, z}]    (* A054088 *)
Intersection[u, v]  (* A352676 *)