Cirno's Summation Game

Cirno has just started learning arithmetic and happily computed the sums from \(1\) all the way up to \(n\). She obtained the sequence:

\[ \{a_i = 1 + 2 + \cdots + i\} \quad \text{for } i = 1,2,\ldots, n \]

To check her calculations, she picks two elements \(v_1\) and \(v_2\) (they may be the same) from the sequence according to the following probability distribution. The probability of selecting the \(i\)th element \(a_i\) is given by:

\[ P(i) = \frac{a_i}{\sum_{k=1}^{n} a_k} = \frac{3i(i+1)}{n(n+1)(n+2)} \]

After selecting \(v_1\) and \(v_2\) (with the two choices independent), she uniformly chooses an integer \(a \in [1, v_1]\) and an integer \(b \in [1, v_2]\). Your task is to compute the overall probability that \(a > b\).

Note: For a given pair of selected elements with values \(v_1\) and \(v_2\), the probability that a uniformly chosen \(a \in [1,v_1]\) is greater than a uniformly chosen \(b \in [1,v_2]\) can be computed as follows:

• If \(v_1 \le v_2\) (which happens when the index \(i \le j\)), then
  \[ P(a>b) = \frac{v_1 - 1}{2 v_2} \] (with the special case when \(v_1=1\), so that \(P(a>b)=0\)).
• If \(v_1 > v_2\) (i.e. \(i > j\)), then
  \[ P(a>b) = \frac{2v_1 - v_2 - 1}{2 v_1} \]

Since \(v_i = \frac{i(i+1)}{2}\), you can substitute to obtain the following cases:

• For \(i < j\):
  \[ P(a>b \mid i,j) = \frac{\frac{i(i+1)}{2} - 1}{j(j+1)} \]
• For \(i = j\):
  \[ P(a>b \mid i,j) = \frac{\frac{i(i+1)}{2} - 1}{i(i+1)} \]
• For \(i > j\):
  \[ P(a>b \mid i,j) = \frac{i(i+1) - \frac{j(j+1)}{2} - 1}{i(i+1)} \]

The overall probability is computed by summing over all pairs \((i,j)\) with the corresponding selection probabilities:

[ \text{Probability} = \sum_{i=1}^{n} \sum_{j=1}^{n} \left(\frac{3i(i+1)}{n(n+1)(n+2)}\right) \left(\frac{3j(j+1)}{n(n+1)(n+2)}\right) \cdot P(a>b \mid i,j). ]

Output the computed probability. It is recommended to use a floating-point output with at least 6 decimal places of precision.

inputFormat

The input consists of a single integer \(n\) (\(1 \le n \le 1000\) for example) on a single line.

outputFormat

Output the probability that \(a > b\) as a floating-point number with at least 6 decimal places.

sample

1

0.000000