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A194734 Number of k such that {-k*r} > {-n*r}, where { } = fractional part, r = (1+sqrt(5))/2 (the golden ratio).

%I A194734 #7 Aug 13 2021 09:37:25
%S A194734 0,0,2,1,0,4,2,7,4,1,8,4,0,9,4,14,8,2,14,7,20,12,4,19,10,1,18,8,26,15,
%T A194734 4,24,12,0,22,9,32,18,4,29,14,40,24,8,36,19,2,32,14,45,26,7,40,20,54,
%U A194734 33,12,48,26,4,42,19,58,34,10,51,26,1,44,18,62,35,8,54,26,73,44
%N A194734 Number of k such that {-k*r} > {-n*r}, where { } = fractional part, r = (1+sqrt(5))/2 (the golden ratio).
%C A194734 The fractional part here uses the Mma implementation for negative arguments: It is the fractional part of the absolute value, turned negative. So a(n) = A019587(n)-1. - _R. J. Mathar_, Aug 13 2021
%t A194734 r = -GoldenRatio; p[x_] := FractionalPart[x];
%t A194734 u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
%t A194734 v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
%t A194734 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A194734 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A194734 Table[s[n], {n, 1, 100}]   (* A019588 *)
%t A194734 Table[t[n], {n, 1, 100}]   (* A193734 *)
%Y A194734 Cf. A019588, A194738.
%K A194734 nonn
%O A194734 1,3
%A A194734 _Clark Kimberling_, Sep 02 2011