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

%I A194733 #24 Aug 16 2022 16:07:57
%S A194733 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,
%T A194733 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,
%U A194733 22,44,9,32,55,18,42,4,29,54,14,40,66,24,51,8,36,64,19,48,2,32
%N A194733 Number of k < n such that {k*r} > {n*r}, where { } = fractional part, r = (1+sqrt(5))/2 (the golden ratio).
%C A194733 The maximum possible value of a(n) is n-1. - _Michael B. Porter_, Jan 29 2012
%H A194733 Reinhard Zumkeller, <a href="/A194733/b194733.txt">Table of n, a(n) for n = 1..1000</a>
%F A194733 a(n)+A019587(n)=n.
%e A194733 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
%p A194733 Digits := 100;
%p A194733 A194733 := proc(n::posint)
%p A194733     local a,k,phi,kfrac,nfrac ;
%p A194733     phi := (1+sqrt(5))/2 ;
%p A194733     a :=0 ;
%p A194733     nfrac := n*phi-floor(n*phi) ;
%p A194733     for k from 1 to n-1 do
%p A194733         kfrac := k*phi-floor(k*phi) ;
%p A194733         if evalf(kfrac-nfrac)  > 0 then
%p A194733             a := a+1 ;
%p A194733         end if;
%p A194733     end do:
%p A194733     a ;
%p A194733 end proc:
%p A194733 seq(A194733(n),n=1..100) ;  # _R. J. Mathar_, Aug 13 2021
%t A194733 r = GoldenRatio; p[x_] := FractionalPart[x];
%t A194733 u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
%t A194733 v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
%t A194733 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A194733 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A194733 Table[s[n], {n, 1, 100}]  (* A019587 *)
%t A194733 Table[t[n], {n, 1, 100}]  (* A194733 *)
%o A194733 (Haskell)
%o A194733 a194733 n = length $ filter (nTau <) $
%o A194733             map (snd . properFraction . (* tau) . fromInteger) [1..n]
%o A194733    where (_, nTau) = properFraction (tau * fromInteger n)
%o A194733          tau = (1 + sqrt 5) / 2
%o A194733 -- _Reinhard Zumkeller_, Jan 28 2012
%Y A194733 Cf. A019587, A194738.
%K A194733 nonn
%O A194733 1,4
%A A194733 _Clark Kimberling_, Sep 02 2011