A194667 - OEIS

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

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

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

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

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

%C n-th row sum gives number of k in [0,1] for which <n*r>+<k*r> > 1; see A194668.

%e First thirteen rows:

%e 1

%e 1 0

%e 0 0 0

%e 1 1 1 1

%e 1 1 0 1 1

%e 1 0 0 1 1 0

%e 0 0 0 1 0 0 0

%e 1 1 1 1 1 1 0 1

%e 1 1 0 1 1 0 0 1 1

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

%e 0 0 0 0 0 0 0 0 0 0 0

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

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

%t r = Sqrt[3]; 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}]  (* A194666 *)

%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    (* A194667 *)

%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}]   (* A194668 *)

%Y Cf. A194668.

%K nonn,tabl

%O 1

%A _Clark Kimberling_, Sep 01 2011