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Table 3 Numerical results of \(\pmb{a_{i}=F_{i}}\) , \(\pmb{r=2}\)

From: The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers

n

Theorem  2.8

Theorem  1.5

\(\frac{\mathbf{Third\ column}}{\mathbf{Second\ column}}\)

2

2

2

\(\frac{2}{2}=1\)

3

4

4

\(\frac{4}{4}=1\)

4

\(6\sqrt{2}\)

12

\(\frac{12}{6\sqrt{2}}=\sqrt{2}\)

5

\(4\sqrt{15}\)

\(\sqrt{30}\)

\(\frac{30}{4\sqrt{15}}\approx 1.936\)

6

\(20\sqrt{2}\)

80

\(\frac{80}{20\sqrt{2}}=2\sqrt{2}\)

n

\(\sqrt{(n-1)|r|^{2}F_{n}F_{n-1}}\)

\(|r|F_{n}F_{n-1}\)

\(\sqrt {(n-1)^{-1}F_{n}F_{n-1}}\)