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%I #11 Feb 09 2018 03:21:08
%S 1,0,0,1,0,1,0,0,1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0,1,0,1,1,1,1,
%T 0,1,0,0,1,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0,0,0
%N Triangular array: T(n,k)=[<n*r>+<k*r>], where [ ] = floor, < > = fractional part, and r=3-sqrt(2).
%C n-th row sum gives number of k in [0,1] for which <n*r>+<k*r> > 1; see A194678.
%H G. C. Greubel, <a href="/A194681/b194681.txt">Table of n, a(n) for the first 150 rows, flattened</a>
%e First ten rows:
%e 1
%e 0 0
%e 1 0 1
%e 0 0 1 0
%e 1 1 1 1 1
%e 1 0 1 0 1 1
%e 0 0 0 0 1 0 0
%e 1 0 1 1 1 1 0 1
%e 0 0 1 0 1 0 0 0 0
%e 1 1 1 1 1 1 0 1 1 1
%t r = 3 - Sqrt[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 (* A194679 *)
%t TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t s[n_] := Sum[w[n, k], {k, 1, n}] (* A194680 *)
%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 (* A194681 *)
%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}] (* A194682 *)
%o (PARI) for(n=1,10, for(k=1,n, print1(floor(frac(n*(3-sqrt(2))) + frac(k*(3-sqrt(2)))), ", "))) \\ _G. C. Greubel_, Feb 08 2018
%Y Cf. A194682.
%K nonn,tabl
%O 1
%A _Clark Kimberling_, Sep 01 2011
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