A194676 Number of k in [1,n] for which + > 1, where < > = fractional part.

1, 1, 0, 2, 2, 2, 6, 8, 1, 4, 1, 9, 4, 3, 4, 6, 14, 1, 19, 7, 8, 2, 4, 19, 7, 21, 22, 10, 2, 10, 10, 24, 2, 30, 27, 7, 11, 7, 38, 14, 8, 13, 14, 37, 0, 4, 12, 15, 3, 15, 4, 35, 38, 15, 26, 5, 45, 53, 17, 17, 4, 1, 12, 60, 19, 3, 20, 20, 50, 5, 48, 22, 47, 66, 0, 75, 73, 27, 25

Offset: 1

Author

Clark Kimberling, Sep 01 2011

Keywords

• nonn

Crossrefs

Cf. A194675.

Programs

Mathematica

r = E; 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}]]
 (* A194675 *)
TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
s[n_] := Sum[w[n, k], {k, 1, n}]  (* A194676 *)
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}]]
  (* A194677 *)
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}]  (* A194678 *)