A134945 Decimal expansion of 1 + sqrt(5).

3, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, 7, 8, 9, 6, 9, 6, 4, 0, 9, 1, 7, 3, 6, 6, 8, 7, 3, 1, 2, 7, 6, 2, 3, 5, 4, 4, 0, 6, 1, 8, 3, 5, 9, 6, 1, 1, 5, 2, 5, 7, 2, 4, 2, 7, 0, 8, 9, 7, 2, 4, 5, 4, 1, 0, 5, 2, 0, 9, 2, 5, 6, 3, 7, 8, 0, 4, 8, 9, 9, 4, 1, 4, 4, 1, 4

Offset: 1

Author

Omar E. Pol, Nov 15 2007

Keywords

• nonn
• cons

Comments

If "index" equals (0,2) then this sequence is the decimal expansion of (golden ratio divided by 5 = phi/5 = (1 + sqrt(5))/10). Example: 0.323606797...

Apart from the leading digit the same as A134972, A098317 and A002163. - R. J. Mathar, Aug 06 2013

Length of the longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020

Abscissa of the first superstable point of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

Examples

			3.2360679774997896964...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Links

• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A000045, A002163, A098317, A134972.

Programs

Mathematica

RealDigits[1 + Sqrt[5], 10, 100][[1]] (* Michael De Vlieger, Nov 13 2020 *)

PARI

1 + sqrt(5) \\ Altug Alkan, Mar 19 2018

Formula

From Christian Katzmann, Mar 19 2018: (Start)

Equals Sum_{n>=0} (15*(2*n)!+8*n!^2)/(n!^2*3^(2*n+2)).

Equals 1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)

Equals 1/A019827. - R. J. Mathar, Jan 17 2021

Equals Product_{k>=1} (1 + 1/Fibonacci(2*k)). - Amiram Eldar, May 27 2021

Extensions

More terms from Jinyuan Wang, Mar 30 2020