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A019588 The right budding sequence: # of i such that 0<i<=n and {tau*n} <= {tau*i} < 1, where {} is fractional part. 4
1, 2, 1, 3, 5, 2, 5, 1, 5, 9, 3, 8, 13, 5, 11, 2, 9, 16, 5, 13, 1, 10, 19, 5, 15, 25, 9, 20, 3, 15, 27, 8, 21, 34, 13, 27, 5, 20, 35, 11, 27, 2, 19, 36, 9, 27, 45, 16, 35, 5, 25, 45, 13, 34, 1, 23, 45, 10, 33, 56, 19, 43, 5, 30, 55, 15, 41, 67, 25, 52, 9, 37, 65, 20, 49, 3, 33, 63, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

J. H. Conway, personal communication.

LINKS

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

N. J. A. Sloane, Classic Sequences

FORMULA

a(n) = A194733(n)+1.

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}]   (* A019588 *)

Table[t[n], {n, 1, 100}]   (* A194734 *)

(* Clark Kimberling, Sep 02 2011 *)

Fold[Join[#1, Range[#1[[#2]], Length[#1] + 1 + Floor[GoldenRatio (#2 + 1)] - Floor[GoldenRatio #2], #2 + 1]] &, {1, 2}, Range[30]] (* Birkas Gyorgy, May 24 2012 *)

PROG

(Haskell)

a019588 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. A019587, A194734, A194738.

Sequence in context: A248408 A210880 A210867 * A193953 A201377 A322942

Adjacent sequences:  A019585 A019586 A019587 * A019589 A019590 A019591

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 December 3 21:17 EST 2021. Contains 349468 sequences. (Running on oeis4.)