A019588 - OEIS

A019588

The right budding sequence: # of i such that 0 < i <= n and {tau*n} <= {tau*i} < 1, where {} is fractional part.

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 A368070

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