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A194813 Number of integers k in [1,n] such that {n*r + k*r} < {n*r - k*r}, where { } = fractional part and r = (1+sqrt(5))/2 (the golden ratio).

%I A194813 #15 Jun 25 2022 10:00:24
%S A194813 0,0,1,2,2,2,3,4,5,5,5,6,6,6,6,7,8,8,8,9,10,11,11,12,13,13,13,13,14,
%T A194813 15,15,15,16,16,16,16,17,18,18,18,19,20,21,21,22,23,23,23,23,24,25,25,
%U A194813 25,26,27,28,28,29,30,30,30,31,32,33,33,33,34,34,34,34,35,36,36
%N A194813 Number of integers k in [1,n] such that {n*r + k*r} < {n*r - k*r}, where { } = fractional part and r = (1+sqrt(5))/2 (the golden ratio).
%C A194813 A194813 + A194814 = A000027 for n > 0.
%e A194813 {4r+1r} = 0.09...; {4r-1r} = 0.85...;
%e A194813 {4r+2r} = 0.70...; {4r-2r} = 0.23...;
%e A194813 {4r+3r} = 0.32...; {4r-3r} = 0.61...;
%e A194813 {4r+4r} = 0.94...; {4r-4r} = 0.00...;
%e A194813 so that a(4)=2.
%t A194813 r = GoldenRatio; p[x_] := FractionalPart[x];
%t A194813 u[n_, k_] := If[p[n*r + k*r] <= p[n*r - k*r], 1, 0]
%t A194813 v[n_, k_] := If[p[n*r + k*r] > p[n*r - k*r], 1, 0]
%t A194813 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A194813 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A194813 Table[s[n], {n, 1, 100}]   (* A194813 *)
%t A194813 Table[t[n], {n, 1, 100}]   (* A194814 *)
%Y A194813 Cf. A001622, A194814, A194738.
%Y A194813 Partial sums of A327174.
%K A194813 nonn
%O A194813 1,4
%A A194813 _Clark Kimberling_, Sep 03 2011