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A019587 The left budding sequence: # of i such that 0<i<=n and 0 < {tau*i} <= {tau*n}, where {} is fractional part. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A019587+A194733=A000027.

REFERENCES

J. H. Conway, personal communication.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

N. J. A. Sloane, Classic Sequences

EXAMPLE

{r}=0.61...; {2r}=0.23...; {3r}=0.85...; {4r}=0.47...;

so that a(4)=2.

MATHEMATICA

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 *)

(* Clark Kimberling, Sep 02 2011 *)

PROG

(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

CROSSREFS

Cf. A019588, A194733, A193738.

Sequence in context: A171746 A113977 A183162 * A102427 A080883 A021315

Adjacent sequences:  A019584 A019585 A019586 * A019588 A019589 A019590

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane and J. H. Conway

EXTENSIONS

Extended by Ray Chandler, Apr 18 2009

STATUS

approved

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Last modified October 18 12:24 EDT 2018. Contains 316321 sequences. (Running on oeis4.)