A194733 Number of k < n such that {k*r} > {n*r}, where { } = fractional part, r = (1+sqrt(5))/2 (the golden ratio).

0, 1, 0, 2, 4, 1, 4, 0, 4, 8, 2, 7, 12, 4, 10, 1, 8, 15, 4, 12, 0, 9, 18, 4, 14, 24, 8, 19, 2, 14, 26, 7, 20, 33, 12, 26, 4, 19, 34, 10, 26, 1, 18, 35, 8, 26, 44, 15, 34, 4, 24, 44, 12, 33, 0, 22, 44, 9, 32, 55, 18, 42, 4, 29, 54, 14, 40, 66, 24, 51, 8, 36, 64, 19, 48, 2, 32

Offset

1,4

Comments

The maximum possible value of a(n) is n-1. - Michael B. Porter, Jan 29 2012

Links

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

Formula

a(n)+A019587(n)=n.

Example

r = 1.618, 2r = 3.236, 3r = 4.854, and 4r = 6.472, where r=(1+sqrt(5))/2.  The fractional part of 4r is 0.472, which is less than the fractional parts of two of {r, 2r, 3r}, so a(4) = 2. - Michael B. Porter, Jan 29 2012

Maple

Digits := 100;
A194733 := proc(n::posint)
    local a, k, phi, kfrac, nfrac ;
    phi := (1+sqrt(5))/2 ;
    a :=0 ;
    nfrac := n*phi-floor(n*phi) ;
    for k from 1 to n-1 do
        kfrac := k*phi-floor(k*phi) ;
        if evalf(kfrac-nfrac)  > 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A194733(n), n=1..100) ;  # R. J. Mathar, Aug 13 2021

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 *)

Prog

(Haskell)
a194733 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, A194738.

Sequence in context: A371767 A157284 A361523 * A143973 A011167 A014176

Adjacent sequences: A194730 A194731 A194732 * A194734 A194735 A194736

Keyword

nonn

Author

Clark Kimberling, Sep 02 2011

Status

approved