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%I #5 Mar 30 2012 18:57:43
%S 0,2,0,1,4,1,4,8,3,7,0,4,10,1,6,13,3,10,18,6,15,1,9,19,3,13,24,7,18,
%T 30,11,24,3,15,29,6,20,35,11,26,0,15,32,4,20,38,9,27,46,15,34,1,20,41,
%U 6,26,48,12,34,57,19,43,3,26,51,9,34,60,17,43,70,25,53,6,33,62
%N Number of k in [1,n] for which <n*r>+<k*r> > 1, where < > = fractional part, and r=(1+sqrt(3))/2; row sums of A164685.
%t r = 1/2 + Sqrt[3]/2; z = 15;
%t p[x_] := FractionalPart[x]; f[x_] := Floor[x];
%t w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]
%t Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t (* A194683 *)
%t TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t s[n_] := Sum[w[n, k], {k, 1, n}]
%t Table[s[n], {n, 1, 100}] (* A194684 *)
%t h[n_, k_] := f[p[n*r] + p[k*r]]
%t Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
%t (* A194685 *)
%t TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
%t t[n_] := Sum[h[n, k], {k, 1, n}]
%t Table[t[n], {n, 1, 100}] (* A194686 *)
%Y Cf. A194684.
%K nonn
%O 1,2
%A _Clark Kimberling_, Sep 01 2011
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