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%I #5 Mar 30 2012 18:57:43
%S 0,2,2,2,5,3,7,1,7,8,11,3,11,12,12,0,15,6,16,19,7,5,21,12,23,0,15,15,
%T 7,15,21,10,23,8,0,24,26,19,10,0,28,29,21,39,0,43,47,24,46,39,11,3,37,
%U 40,3,49,26,51,41,3,52,23,16,57,4,3,43,45,27,13,26,68,50,49,37
%N Number of k in [1,n] for which <r^n>+<r^k> > 1, where < > = fractional part, and r=(1+sqrt(3))/2.
%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. A194683.
%K nonn
%O 1,2
%A _Clark Kimberling_, Sep 01 2011
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