A194669 Number of k in [1,n] for which + > 1, where < > = fractional part and r = sqrt(5).

0, 0, 0, 0, 3, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7, 0, 0, 0, 2, 0, 3, 0, 8, 0, 8, 0, 13, 0, 3, 0, 2, 0, 14, 0, 0, 0, 0, 0, 6, 0, 16, 0, 21, 0, 20, 0, 9, 0, 14, 0, 18, 0, 16, 0, 4, 0, 26, 0, 11, 0, 18, 0, 5, 0, 30, 0, 20, 0, 32, 0, 21, 0, 21, 0, 20, 0, 13, 0, 20, 0, 13, 0, 30, 0, 19, 0, 10, 0

Offset: 1

Author

Clark Kimberling, Sep 01 2011

Keywords

• nonn

Programs

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 *)