A329827 Beatty sequence for (5+sqrt(37))/6.

1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 36, 38, 40, 42, 44, 46, 48, 49, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 70, 72, 73, 75, 77, 79, 81, 83, 84, 86, 88, 90, 92, 94, 96, 97, 99, 101, 103, 105, 107, 108, 110, 112, 114, 116

Offset: 1

Author

Clark Kimberling, Dec 31 2019

Keywords

• nonn
• easy

Comments

Let r = (5+sqrt(37))/6. Then (floor(n*r)) and (floor(n*r + r/3)) 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, A329828 (complement).

Programs

Mathematica

t = 1/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329827 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329828 *)

Formula

a(n) = floor(n*r), where r = (5+sqrt(37))/6.