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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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4
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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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See A190508.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Weiru Chen, Jared Krandel, Interpolating Classical Partitions of the Set of Positive Integers, arXiv:1810.11938 [math.NT], 2018. See sequence D1 p. 4.
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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.
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
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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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(See A190508.)
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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(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
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CROSSREFS
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Cf. A054770, A190508, A190511.
Sequence in context: A288316 A003250 A320494 * A022131 A091391 A135105
Adjacent sequences: A190506 A190507 A190508 * A190510 A190511 A190512
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, May 11 2011
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STATUS
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approved
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