n

$\sqrt{n}=x_1+x_1^{-1}$

notation

$x_1$

notation

$x_1^{-1}$

4

2

${\phi }^0$

1

${\phi }^0$

1

5

$\sqrt{5}$

${\phi }^{-1}$

1.6180339887

$\phi$

0.6180339887

8

$2\cdot \sqrt{2}$

${\delta }_s^{-1}$

2.4142135623

${\delta }_s$

0.414213562

9

3

${\phi }^{-2}$

2.6180339887

${\phi }^2$

0.381966011

18

4² + 2

$3\cdot \sqrt{2}$

3.9921490369

20

$2\cdot \sqrt{5}$

${\phi }^{-3}$

4.2360679774

${\phi }^3$

0.236067977

27

5² + 2

4.9959919730

32

$4\cdot \sqrt{2}$

5.4741784358

0.1826758136

38

6² + 2

5.9976829489

49

7

${\phi }^{-4}$

6.8541019662

${\phi }^4$

0.1458980337

51

7² + 2

6.9985415144

66

8² + 2

7.9990231393

83

9² + 2

8.9993139982

102

10² + 2

9.9994999374

123

11² + 2

10.999624310

125

$5\cdot \sqrt{5}$

${\phi }^{-5}$

11.090169943

${\phi }^5$

0.090169943

146

12² + 2

11.999710630

171

13² + 2

12.999772406