A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

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

Offset: 1

Author

Alonso del Arte, Mar 17 2012

Keywords

• nonn
• cons
• easy

Comments

The number x given by the infinitely nested radical for n = 3 is such that x^2 = x + 3, bearing some similarity to the golden ratio phi with its property that phi^2 = phi + 1. Also, 3/x = x - 1.

The mentioned polynomial x^2 - x - 3 has the present number as positive root, and the negative one is -A223139. - Wolfdieter Lang, Aug 29 2022

It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023

c^n = A006130(n) + A006130(n-1) * d, where c = (1 + sqrt(13))/2 and d = (-1 + sqrt(13))/2. - Gary W. Adamson, Nov 25 2023

c^n = A052533(n) + A006130(n-1)*c, with A006130(-1) = 0. This is also valid for powers of 1/c = A356033, with A052533 and A006130 given there in terms of S-Chebyshev polynomials (A049310), used for negative indices. - Wolfdieter Lang, Nov 26 2023

Examples

			2.30277563773...
		

Links

• Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 274.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A157532 (continued fraction), A001622, A010470, A085550, A098316, A188943, A223139.

Cf. A006130, A049310, A052533, A209927.

Programs

Maple

Digits:=140:
evalf((sqrt(13)+1)/2);  # Alois P. Heinz, Sep 19 2023

Mathematica

RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]
RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* Robert G. Wilson v, Oct 02 2018 *)

PARI

(sqrt(13)+1)/2 \\ Altug Alkan, Oct 03 2018

Formula

Closed form: (sqrt(13) + 1)/2 = A098316-1 = A085550+2 = 3*(A188943-1).