login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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