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A194681
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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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4
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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, 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, 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
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OFFSET
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1
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COMMENTS
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n-th row sum gives number of k in [0,1] for which <n*r>+<k*r> > 1; see A194678.
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LINKS
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EXAMPLE
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First ten rows:
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 0 1
0 0 1 0 1 0 0 0 0
1 1 1 1 1 1 0 1 1 1
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MATHEMATICA
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r = 3 - Sqrt[2]; z = 15;
p[x_] := FractionalPart[x]; f[x_] := Floor[x];
w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]
Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
s[n_] := Sum[w[n, k], {k, 1, n}] (* A194680 *)
Table[s[n], {n, 1, 100}]
h[n_, k_] := f[p[n*r] + p[k*r]]
Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
t[n_] := Sum[h[n, k], {k, 1, n}]
Table[t[n], {n, 1, 100}] (* A194682 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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