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%I #5 Mar 30 2012 18:57:43
%S 0,0,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0,0,1,1,1,0,1,1,1,1,1,1,1,1,
%T 1,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,0,1,1,1,0,1,1,1,0,0,1,1,1,
%U 1,1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1
%N Triangular array: T(n,k)=[<n*r>+<k*r>], where [ ] = floor, < > = fractional part, and r = sqrt(5).
%C n-th row sum gives number of k in [0,1] for which <n*r>+<k*r> > 1; see A194671.
%e First thirteen rows:
%e 0
%e 0 0
%e 0 1 1
%e 1 1 1 1
%e 0 0 0 1 0
%e 0 0 1 1 0 0
%e 0 1 1 1 0 1 1
%e 1 1 1 1 1 1 1 1
%e 0 0 0 1 0 0 0 1 0
%e 0 0 1 1 0 0 1 1 0 0
%e 0 1 1 1 0 1 1 1 0 0 1
%e 1 1 1 1 1 1 1 1 0 1 1 1
%e 0 0 0 1 0 0 0 0 0 0 0 0 0
%t r = Sqrt[5]; z = 13;
%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 TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t s[n_] := Sum[w[n, k], {k, 1, n}] (* A194669 *)
%t Table[s[n], {n, 1, 100}]
%t h[n_, k_] := f[p[n*r] + p[k*r]]
%t Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
%t (* A194670 *)
%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}] (* A194671 *)
%Y Cf. A194671.
%K nonn,tabl
%O 1
%A _Clark Kimberling_, Sep 01 2011
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