import java.math.BigInteger;

public class Euler825 {

    static double digamma(double x) {
        if (x <= 0.0 || Double.isInfinite(x) || Double.isNaN(x)) {
            return Double.NaN;
        }

        double res = 0.0;
        while (x < 10.0) {
            res -= 1.0 / x;
            x += 1.0;
        }

        double inv = 1.0 / x;
        double inv2 = inv * inv;

        double series = inv2 * (-1.0 / 12.0
                + inv2 * (1.0 / 120.0
                        + inv2 * (-1.0 / 252.0
                                + inv2 * (1.0 / 240.0
                                        + inv2 * (-1.0 / 132.0
                                                + inv2 * (691.0 / 32760.0))))));

        res += Math.log(x) - 0.5 * inv + series;
        return res;
    }

    public static String solve() {
        long N = 100000000000000L;
        double sqrt3 = Math.sqrt(3.0);
        double c = (3.0 + sqrt3) / 6.0;

        int K = 60;
        BigInteger[] num = new BigInteger[K + 1];
        BigInteger[] den = new BigInteger[K + 1];

        for (int i = 0; i <= K; i++) {
            num[i] = BigInteger.ZERO;
            den[i] = BigInteger.ZERO;
        }

        num[2] = BigInteger.valueOf(7);
        num[3] = BigInteger.valueOf(35);
        num[4] = BigInteger.valueOf(121);

        den[2] = BigInteger.valueOf(11);
        den[3] = BigInteger.valueOf(73);
        den[4] = BigInteger.valueOf(395);
        den[5] = BigInteger.valueOf(1933);

        for (int n = 5; n <= K; ++n) {
            num[n] = num[n - 1].multiply(BigInteger.valueOf(3))
                    .add(num[n - 2].multiply(BigInteger.valueOf(3)))
                    .subtract(num[n - 3]);
        }
        for (int n = 6; n <= K; ++n) {
            den[n] = den[n - 1].multiply(BigInteger.valueOf(8))
                    .subtract(den[n - 2].multiply(BigInteger.valueOf(18)))
                    .add(den[n - 3].multiply(BigInteger.valueOf(8)))
                    .subtract(den[n - 4]);
        }

        double E = 0.0;
        for (int n = 2; n <= K; ++n) {
            // Convert to double. Alternatively we can use BigDecimal
            double nDouble = num[n].doubleValue();
            double dDouble = den[n].doubleValue();
            double s = nDouble / dDouble;
            E += s - 1.0 / (n - c);
        }

        double xBig = (double) N + 1.0 - c;
        double xSmall = 2.0 - c;

        double ans = digamma(xBig) - digamma(xSmall) + E;

        return String.format(java.util.Locale.US, "%.8f", ans);
    }

    public static void main(String[] args) {
        System.out.println(solve());
    }
}