A194683 Triangular array: T(n,k)=[<r^n>+<r^k>], where [ ] = floor, < > = fractional part, and r=(1+sqrt(3))/2.

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, 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, 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

Offset

1

Comments

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

Links

Table of n, a(n) for n=1..99.

Example

First ten rows:
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 1 0
0 1 1 1 1 1 1 0 1
0 1 1 1 1 1 1 0 1 1

Mathematica

r = 1/2 + Sqrt[3]/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}]]
    (* A194683 *)
TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
s[n_] := Sum[w[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194684 *)
h[n_, k_] := f[p[n*r] + p[k*r]]
Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
 (* A194685 *)
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}]    (* A194686 *)

Crossrefs

Cf. A194684.

Sequence in context: A022928 A000494 A022933 * A306710 A188295 A228039

Adjacent sequences: A194680 A194681 A194682 * A194684 A194685 A194686

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 01 2011

Status

approved