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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 11 2011
STATUS
approved