A190509 a(n) = n + [nr/s] + [nt/s] + [nu/s] where r=golden ratio, s=r^2, t=r^3, u=r^4, and [] represents the floor function.

4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 69, 76, 80, 87, 91, 98, 105, 109, 116, 120, 127, 134, 138, 145, 152, 156, 163, 167, 174, 181, 185, 192, 199, 203, 210, 214, 221, 228, 232, 239, 243, 250, 257, 261, 268, 275, 279, 286, 290, 297, 304, 308, 315, 319, 326, 333, 337, 344, 351, 355, 362, 366, 373, 380, 384, 391, 398, 402, 409

Offset

1,1

Comments

See A190508.

From Clark Kimberling, Nov 13 2022: (Start)

This is the third of four sequences that partition the positive integers. 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 A356104 to A356107.

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 this sequence, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

(1) v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217

(2) v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218

(3) v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = this sequence

(4) v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Weiru Chen and Jared Krandel, Interpolating Classical Partitions of the Set of Positive Integers, arXiv:1810.11938 [math.NT], 2018. See sequence D1 p. 4. Also in The Ramanujan Journal, (2020).

Formula

A190508: a(n) = n + [nr] + [nr^2] + [nr^3];

A190509: b(n) = [n/r] + n + [nr] + [nr^2];

A054770: c(n) = [n/r^2] + [n/r] + n + [nr];

A190511: d(n) = [n/r^3] + [n/r^2] + [n/r] + n.

a(n) = 3*A000201(n)+n, since r/s = 1/r = r-1, and u/s = r^2 = r+1. - Michel Dekking, Sep 06 2017

a(n) = A000201(n) + A003623(n). - Primoz Pirnat, Jan 08 2021

Maple

r:=(1+sqrt(5))/2: s:=r^2: t:=r^3: u:=r^4: a:=n->n+floor(n*r/s)+floor(n*t/s)+floor(n*u/s):  seq(a(n), n=1..70); # Muniru A Asiru, Nov 01 2018

Mathematica

(See A190508.)
Table[3 Floor[n (Sqrt[5] + 1) / 2] + n, {n, 1, 100}] (* Vincenzo Librandi, Nov 01 2018 *)

Prog

(PARI) a(n) = 3*floor(n*(sqrt(5)+1)/2) + n; \\ Michel Marcus, Sep 10 2017; after Michel Dekking's formula
(Magma) [3*Floor(n*(Sqrt(5)+1)/2) + n: n in [1..80]]; // Vincenzo Librandi, Nov 01 2018
(Python)
from math import isqrt
def A190509(n): return n+((m:=n+isqrt(5*n**2))&-2)+(m>>1) # Chai Wah Wu, Aug 10 2022

Crossrefs

Cf. A054770, A190508, A190511.

Cf. A000201, A001622, A003623, A340429.

Sequence in context: A288316 A003250 A320494 * A022131 A091391 A135105

Adjacent sequences: A190506 A190507 A190508 * A190510 A190511 A190512

Keyword

nonn

Author

Clark Kimberling, May 11 2011

Status

approved