%I #44 Dec 16 2023 04:50:54

%S 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,

%T 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,

%U 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

%N Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

%C 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.

%C 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

%C It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - _Stefano Spezia_, Sep 19 2023

%C 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 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

%H Alberto Seeger and David Sossa, <a href="https://ajc.maths.uq.edu.au/pdf/87/ajc_v87_p258.pdf">Infinite families of connected graphs with equal spectral radius</a>, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 274.

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

%e 2.30277563773...

%p Digits:=140:

%p evalf((sqrt(13)+1)/2); # _Alois P. Heinz_, Sep 19 2023

%t RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]

%t RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 02 2018 *)

%o (PARI) (sqrt(13)+1)/2 \\ _Altug Alkan_, Oct 03 2018

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

%Y Cf. A006130, A049310, A052533, A209927.

%K nonn,cons,easy

%O 1,1

%A _Alonso del Arte_, Mar 17 2012