A296184 Decimal expansion of 2 + phi, with the golden section phi from A001622.

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

Offset: 1

Author

Wolfdieter Lang, Jan 08 2018

Keywords

• nonn
• cons
• easy

Comments

In a regular pentagon, inscribed in a unit circle this equals twice the largest distance between a vertex and a midpoint of a side.

This is an integer in the quadratic number field Q(sqrt(5)).

Only the first digit differs from A001622.

Examples

			3.618033988749894848204586834365638117720309179805762862135448622705260462...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.25, p. 417.

Links

• Sumit Kumar Jha, Two complementary relations for the Rogers-Ramanujan continued fraction, arXiv:2112.12081 [math.NT], 2021.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A001622, A094214, A104457, A176055, A020837.

2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A332438 (n = 9), A019973 (n = 12).

Programs

Mathematica

First@ RealDigits[2 + GoldenRatio, 10, 77] (* Michael De Vlieger, Jan 13 2018 *)

PARI

(5 + sqrt(5))/2 \\ Altug Alkan, Mar 19 2018

Formula

Equals 2 + A001622 = 1 + A104457 = 3 + A094214.

From Christian Katzmann, Mar 19 2018: (Start)

Equals Sum_{n>=0} (15*(2*n)!+40*n!^2)/(2*n!^2*3^(2*n+2)).

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

Constant c = 2 + 2*cos(2*Pi/10). The linear fractional transformation z -> c - c/z has order 10, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z)))))))))). - Peter Bala, May 09 2024