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A078588
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a(n) = 1 if the integer multiple of phi nearest n is greater than n, otherwise 0, where phi = (1+sqrt(5))/2.
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27
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0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1
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OFFSET
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0,1
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COMMENTS
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From Fred Lunnon, Jun 20 2008: (Start)
Partition the positive integers into two sets A_0 and A_1 defined by A_k == { n | a(n) = k }; so A_0 = A005653 = { 2, 4, 5, 7, 10, 12, 13, 15, 18, 20, ... }, A_1 = A005652 = { 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 21, ... }.
Then form the sets of sums of pairs of distinct elements from each set and take the complement of their union: this is the Fibonacci numbers { 1, 2, 3, 5, 8, 13, 21, 34, 55, ... } (see the Chow article). (End)
The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
This is the complement of A089809; also a(n) = 1 iff A024569(n) = 1. - Gary W. Adamson, Nov 11 2003
Since (n*phi) is equidistributed, s(n):=(Sum_{k=1..n}a(k))/n converges to 1/2, but actually s(n) is exactly equal to 1/2 for many values of n. These values are given by A194402. - Michel Dekking, Sep 30 2016
From Clark Kimberling and Jianing Song, Sep 09 2019: (Start)
Suppose that k >= 2, and let a(n) = floor(n*k*r) - k*floor(n*r) = k*{n*r} - {n*k*r}, an integer strictly between 0 and k, where {} denotes fractional part. For h = 0,1,...,k-1, let s(h) be the sequence of positions of h in {a(n)}. The sets s(h) partition the positive integers. Although a(n)/n -> k, the sequence a(n)-k*n appears to be unbounded.
Guide to related sequences, for k = 2:
** r ********* {a(n)} positions of 0's positions of 1's
(1+sqrt(5))/2 A078588 A005653 A005652
sqrt(2) A197879 A120243 A120749
sqrt(3) A190669 A190670 A190671
e A190843 A190847 A190860
Pi A191153 A191159 A191164
sqrt(5) A188257 A188258 A188259
sqrt(6) A327253 A327254 A327255
sqrt(8) A327256 A327257 A327258
Guide to related sequences, for k = 3:
** r ********* {a(n)} pos. of 0's pos. of 1's pos. of 2's
(1+sqrt(5))/2 A140397 A140398 A140399 A140400
sqrt(2) A190487 A190488 A190489 A190490
sqrt(3) A190676 A190677 A190678 A190679
e A190893 A191103 A191104 A191105
Pi A327298 A327299 A327300 A327301
sqrt(5) A189463 A189464 A189465 A190158
sqrt(6) A327306 A327307 A327308 A327309
sqrt(8) A327310 A327311 A327312 A327313
Guide to related sequences, for k = 4:
** r ********* {a(n)} pos. of 0's pos. of 1's pos. of 2's pos. of 3's
sqrt(2) A190544 A190545 A190546 A190547 A190548
sqrt(5) A189480 A190813 A190883 A190884 A190885
(End)
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REFERENCES
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D. L. Silverman, J. Recr. Math. 9 (4) 208, problem 567 (1976-77).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211.
T. Chow, A new characterization of the Fibonacci-free partition, Fibonacci Q. 29 (1991), 174-180; also online here
T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653].
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FORMULA
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a(n) = floor(2*phi*n) - 2*floor(phi*n) where phi denotes the golden ratio (1 + sqrt(5))/2. - Fred Lunnon, Jun 20 2008
a(n) = 2{n*phi} - {2n*phi}, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007
a(n) = n + 1 + ceiling(n*sqrt(5)) - 2*ceiling(n*phi) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Dec 05 2002
a(n) = round(phi*n) - floor(phi*n). - Michel Dekking, Sep 30 2016
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MATHEMATICA
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f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Table[ f[n], {n, 0, 105}]
r = (1 + Sqrt[5])/2; z = 300;
t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 0, z}]
(* Clark Kimberling, 26 Aug 2019 *)
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PROG
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(PARI) a(n)=if(n, n+1+ceil(n*sqrt(5))-2*ceil(n*(1+sqrt(5))/2), 0) \\ (changed by Jianing Song, Sep 10 2019 to include a(0) = 0)
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CROSSREFS
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Cf. A005652, A005653, A024569, A089808, A089809, A140397, A140398, A140399, A140400, A140401.
Sequence in context: A197879 A296657 A082848 * A175992 A173922 A213676
Adjacent sequences: A078585 A078586 A078587 * A078589 A078590 A078591
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KEYWORD
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easy,nonn
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AUTHOR
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Robert G. Wilson v, Dec 02 2002
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EXTENSIONS
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Edited by N. J. A. Sloane, Jun 20 2008, at the suggestion of Fred Lunnon
Edited by Jianing Song, Sep 09 2019
Offset corrected by Jianing Song, Sep 10 2019
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STATUS
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approved
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