A329840 Beatty sequence for (9+sqrt(41))/4.

3, 7, 11, 15, 19, 23, 26, 30, 34, 38, 42, 46, 50, 53, 57, 61, 65, 69, 73, 77, 80, 84, 88, 92, 96, 100, 103, 107, 111, 115, 119, 123, 127, 130, 134, 138, 142, 146, 150, 154, 157, 161, 165, 169, 173, 177, 180, 184, 188, 192, 196, 200, 204, 207, 211, 215, 219

Offset: 1

Author

Clark Kimberling, Dec 31 2019

Keywords

• nonn
• easy

Comments

Let r = (-1+sqrt(41))/4. Then (floor(n*r)) and (floor(n*r + 5r/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

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Beatty Sequence
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A329825, A329839 (complement).

Programs

Mathematica

t = 5/2; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329839 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329840 *)

Formula

a(n) = floor(n*s), where s = (9+sqrt(41))/4. - corrected by Michael De Vlieger, Aug 27 2021