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 A078588 a(n) = 1 if the integer multiple of phi nearest n is greater than n, otherwise 0, where phi = (1+sqrt(5))/2. 27
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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) REFERENCES D. L. Silverman, J. Recr. Math. 9 (4) 208, problem 567 (1976-77). LINKS 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]. FORMULA 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 MATHEMATICA 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 *) PROG (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) CROSSREFS 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 KEYWORD easy,nonn AUTHOR Robert G. Wilson v, Dec 02 2002 EXTENSIONS 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 STATUS approved

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Last modified December 6 07:11 EST 2019. Contains 329785 sequences. (Running on oeis4.)