A222132 Decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))).

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

Offset: 1

Author

Jaroslav Krizek, Feb 08 2013

Keywords

• nonn
• cons
• easy

Comments

Sequence with a(1) = 1 is decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) = A222133.

Because 17 == 1 (mod 4), the basis for integers in the real quadratic number field K(sqrt(17)) is <1, omega(17)>, where omega(17) = (1 + sqrt(17))/2. - Wolfdieter Lang, Feb 10 2020

This is the positive root of the polynomial x^2 - x - 4, with negative root -A222133. - Wolfdieter Lang, Dec 10 2022

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

c^n = A006131(n) + A006131(n-1) * d, where c = (1 + sqrt(17))/2 and d = (-1 + sqrt(17))/2. - Gary W. Adamson, Nov 25 2023

c^n = A052923(n) + A006131(n-1) * c. Also for negative n. - Wolfdieter Lang, Nov 27 2023

The effective degree of maximal entropy random walk on the barred-square graph (see Burda et al.). - Stefano Spezia, Feb 07 2025

Examples

			2.561552812808830274910704...
		

Links

• J. Burda, J. Duda, J. M. Luck, and B. Waclaw, The Various Facets of Random Walk Entropy, Acta Phys. Pol. B 41, 949-987 (2010). See pp. 964, 969.
• Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See pp. 260 and 263.

Crossrefs

Cf. A222133, A178255, A082486.

Cf. A006131, A052923.

Programs

Maple

Digits:=140:
evalf((sqrt(17)+1)/2);  # Alois P. Heinz, Sep 19 2023

Mathematica

RealDigits[(1 + Sqrt[17])/2, 10, 130]

Formula

Closed form: (sqrt(17) + 1)/2 = A178255 - 1 = A082486 - 2.

sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) - 1 = sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))). See A222133.