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Triangular array: T(n,k)=[<n*r>+<k*r>], where [ ] = floor, < > = fractional part, and r=3-sqrt(2).
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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