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Triangular array: T(n,k)=[<r^n>+<r^k>], where [ ] = floor, < > = fractional part, and r=(1+sqrt(3))/2.
5

%I #5 Mar 30 2012 18:57:43

%S 0,1,1,0,1,1,0,1,1,0,1,1,1,1,1,0,1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,

%T 1,0,0,1,1,1,1,1,1,0,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,

%U 0,0,0,0,1,0,0,0,1,0,1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,1,1,1,1,1,0

%N Triangular array: T(n,k)=[<r^n>+<r^k>], where [ ] = floor, < > = fractional part, and r=(1+sqrt(3))/2.

%C n-th row sum gives number of k in [0,1] for which <r^n>+<r^k> > 1; see A194684.

%e First ten rows:

%e 0

%e 1 1

%e 0 1 1

%e 0 1 1 0

%e 1 1 1 1 1

%e 0 1 1 0 1 0

%e 1 1 1 1 1 1 1

%e 0 0 0 0 0 0 1 0

%e 0 1 1 1 1 1 1 0 1

%e 0 1 1 1 1 1 1 0 1 1

%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. A194684.

%K nonn,tabl

%O 1

%A _Clark Kimberling_, Sep 01 2011