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A190509
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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.
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12
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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
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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FORMULA
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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.
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MAPLE
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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
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MATHEMATICA
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Table[3 Floor[n (Sqrt[5] + 1) / 2] + n, {n, 1, 100}] (* Vincenzo Librandi, Nov 01 2018 *)
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PROG
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(Magma) [3*Floor(n*(Sqrt(5)+1)/2) + n: n in [1..80]]; // Vincenzo Librandi, Nov 01 2018
(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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