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

Offset: 1

Author

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

Keywords

• nonn
• easy
• nice

Comments

Also, the number of distinct blocks of the Fibonacci word (A003849) containing the maximum possible number of 1's for such a block. - Jeffrey Shallit, Jul 09 2025

References

• J. H. Conway, personal communication.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Classic Sequences

Crossrefs

Cf. A003849, A019587, A194734, A194738.

Programs

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

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 *)

Formula

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

Extensions

Extended by Ray Chandler, Apr 18 2009