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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..77.

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.

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

Example

3.618033988749894848204586834365638117720309179805762862135448622705260462...

Mathematica

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

Prog

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

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).

Sequence in context: A340310 A096602 A288853 * A290481 A259501 A118948

Adjacent sequences: A296181 A296182 A296183 * A296185 A296186 A296187

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Jan 08 2018

Status

approved