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A019587
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The left budding sequence: # of i such that 0<i<=n and 0 < {tau*i} <= {tau*n}, where {} is fractional part.
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6
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1, 1, 3, 2, 1, 5, 3, 8, 5, 2, 9, 5, 1, 10, 5, 15, 9, 3, 15, 8, 21, 13, 5, 20, 11, 2, 19, 9, 27, 16, 5, 25, 13, 1, 23, 10, 33, 19, 5, 30, 15, 41, 25, 9, 37, 20, 3, 33, 15, 46, 27, 8, 41, 21, 55, 34, 13, 49, 27, 5, 43, 20, 59, 35, 11, 52, 27, 2, 45, 19, 63, 36, 9, 55, 27, 74, 45, 16, 65
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OFFSET
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1,3
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COMMENTS
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A019587+A194733=A000027.
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REFERENCES
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J. H. Conway, personal communication.
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Classic Sequences
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EXAMPLE
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{r}=0.61...; {2r}=0.23...; {3r}=0.85...; {4r}=0.47...;
so that a(4)=2.
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MATHEMATICA
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r = GoldenRatio; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A019587 *)
Table[t[n], {n, 1, 100}] (* A194733 *)
(* from Clark Kimberling, Sep 2 2011 *)
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PROG
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(Haskell)
a019587 n = length $ filter (<= nTau) $
map (snd . properFraction . (* tau) . fromInteger) [1..n]
where (_, nTau) = properFraction (tau * fromInteger n)
tau = (1 + sqrt 5) / 2
-- Reinhard Zumkeller, Jan 28 2012
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CROSSREFS
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Cf. A019588, A194733, A193738.
Sequence in context: A171746 A113977 A183162 * A102427 A080883 A021315
Adjacent sequences: A019584 A019585 A019586 * A019588 A019589 A019590
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane and J. H. Conway (conway(AT)math.princeton.edu)
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EXTENSIONS
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Extended by Ray Chandler, Apr 18 2009
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STATUS
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approved
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