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Search: id:A078588
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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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+0
12
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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 (qntmpkt(AT)yahoo.com), Nov 11 2003
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REFERENCES
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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; online at http://www-math.mit.edu/~tchow/cv.html
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=1..1000
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 + sqrt5)/2. - W. F. Lunnon (fred.lunnon(AT)gmail.com), Jun 20 2008
a(n) = 2{n*phi} - {2n*phi}, where { } denotes fractional part. - Clark Kimberling (ck6(AT)evansville.edu), Jan 01 2007
a(n) = n+1+ceil(n*sqrt(5)) - 2*ceil(n*phi) where phi = (1+sqrt(5))/2. - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 05 2002
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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, 1, 105}]
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PROGRAM
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(PARI) a(n)=n+1+ceil(n*sqrt(5))-2*ceil(n*(1+sqrt(5))/2)
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CROSSREFS
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Cf. A005652, A005653, A024569, A089808, A089809, A140397-A140401.
Sequence in context: A030658 A112539 A104104 this_sequence A039983 A152490 A145273
Adjacent sequences: A078585 A078586 A078587 this_sequence 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 (rgwv(AT)rgwv.com), Dec 02 2002
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Jun 20 2008, at the suggestion of Fred Lunnon.
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