A341239 a(n) = floor(r*floor(s*n)), where r = 1 + sqrt(2) and s = sqrt(2).

2, 4, 9, 12, 16, 19, 21, 26, 28, 33, 36, 38, 43, 45, 50, 53, 57, 60, 62, 67, 70, 74, 77, 79, 84, 86, 91, 94, 98, 101, 103, 108, 111, 115, 118, 120, 125, 127, 132, 135, 137, 142, 144, 149, 152, 156, 159, 161, 166, 168, 173, 176, 178, 183, 185, 190, 193, 197

Offset: 1

Author

Clark Kimberling, Feb 07 2021

Keywords

• nonn

Comments

Conjecture: 1 < r*s*n - a(n) < 3 for n >= 1.

From Clark Kimberling, Dec 27 2022: (Start)

This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:

(1) v o u, defined by (v o u)(n) = v(u(n));

(2) u o v';

(3) v o u';

(4) v' o u'.

Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A184922.

Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and

1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.

For A341239, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).

(1) v o u = (2, 4, 9, 12, 16, 19, 21, 26, 28, 33, 36, 38, ...) = A341239

(2) v' o u = (1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 25, 27, ...) = A286666

(3) v o u' = (7, 14, 24, 31, 41, 48, 55, 65, 72, 82, 89, ...) = A188383

(4) v' o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 63, ...) = A098021

(End)

Crossrefs

Cf. A184922, A341240; A098021, A188383, A341239, A286666.

Programs

Mathematica

z = 140; r = 1 + Sqrt[2]; s = Sqrt[2]; f[x_] := Floor[r*Floor[s*x]];
Table[f[n], {n, 1, z}]

Formula

a(n) = floor(r*floor(s*n)), where r = 1 + sqrt(2) and s = sqrt(2).