A194670 Triangular array: T(n,k)=[<n*r>+<k*r>], where [ ] = floor, < > = fractional part, and r = sqrt(5).

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1

Offset

1

Comments

n-th row sum gives number of k in [0,1] for which <n*r>+<k*r> > 1; see A194671.

Links

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

Example

First thirteen rows:
0
0 0
0 1 1
1 1 1 1
0 0 0 1 0
0 0 1 1 0 0
0 1 1 1 0 1 1
1 1 1 1 1 1 1 1
0 0 0 1 0 0 0 1 0
0 0 1 1 0 0 1 1 0 0
0 1 1 1 0 1 1 1 0 0 1
1 1 1 1 1 1 1 1 0 1 1 1
0 0 0 1 0 0 0 0 0 0 0 0 0

Mathematica

r = Sqrt[5]; z = 13;
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}]  (* A194669 *)
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}]]
(* A194670 *)
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}]  (* A194671 *)

Crossrefs

Cf. A194671.

Sequence in context: A091247 A085137 A304577 * A130543 A185013 A346459

Adjacent sequences: A194667 A194668 A194669 * A194671 A194672 A194673

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 01 2011

Status

approved