A222132 -id:A222132 - cp's OEIS frontend

A365814 Decimal expansion of the largest root of the polynomial x^3 - 2x^2 - 2x + 2.

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

Offset: 1

Stefano Spezia, Sep 19 2023

It is the spectral radius of the house graph.

			2.4811943040920156226335372412168571805527452...

Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 270.
Eric Weisstein's World of Mathematics, House Graph

Cf. A078564, A209927, A222132, A365750.

Digits:= 140:
fsolve(x^3-2*x^2-2*x+2)[3];  # Alois P. Heinz, Sep 19 2023

First[RealDigits[Root[#^3-2#^2-2#+2,3,0],10,88]]

Equals (2 + 10/(3*i*sqrt(111) - 1)^(1/3) + (3*i*sqrt(111) - 1)^(1/3))/3, where i denotes the imaginary unit.

A380895 Decimal expansion of (sqrt(17) + 1)/(4*sqrt(17)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 07 2025

This constant and A380896 give the stationary distribution for maximal entropy random walk on the barred-square graph (see Burda et al.).

			0.310633906259083243379726615529030544487458812...

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 p. 969.
Index entries for algebraic numbers, degree 2.

Cf. A010473, A222132, A380896.

RealDigits[(Sqrt[17]+1)/(4Sqrt[17]),10,100][[1]]

1/4+1/sqrt(272) \\ Charles R Greathouse IV, Feb 08 2025

Equals 1/2 - A380896.

Minimal polynomial: 34*x^2 - 17*x + 2. - Stefano Spezia, Aug 03 2025

A380896 Decimal expansion of (sqrt(17) - 1)/(4*sqrt(17)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 07 2025

A380895 and this constant give the stationary distribution for maximal entropy random walk on the barred-square graph (see Burda et al.).

			0.18936609374091675662027338447096945551254118786...

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 p. 969.
Index entries for algebraic numbers, degree 2.

Cf. A010473, A222132, A380895.

RealDigits[(Sqrt[17]-1)/(4Sqrt[17]),10,100][[1]]

1/4-1/sqrt(272) \\ Charles R Greathouse IV, Feb 08 2025

Equals 1/2 - A380895.

Minimal polynomial: 34*x^2 - 17*x + 2. - Stefano Spezia, Aug 03 2025

A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 03 2018

For x >= 0, sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)))) = (sqrt(4*x+1) + 1)/2. This is an integer for each x such that 2*x is a term in A000217.

			3.541381265149109844499842122601033531042485047393205593209576523243166362659...

Index entries for algebraic numbers, degree 2

Cf. A000007, A001622, A000038, A209927, A222132, A222134, A223140, A235162.

evalf((sqrt(37)+1)/2,120); # Muniru A Asiru, Oct 07 2018

RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[9, {135}]], 10, 111][[1]] (* or *)
RealDigits[(Sqrt[37] + 1)/2, 10, 111][[1]]

(sqrt(37)+1)/2 \\ Altug Alkan, Oct 03 2018

Minimal polynomial: x^2 - x - 9. - Stefano Spezia, Jul 02 2025

A358945 Decimal expansion of the positive root of 4*x^2 + x - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jan 20 2023

The negative root is -(A189038 - 1) = -0.6403882032... .

c^n = A052923(-n) + A006131(-(n+1))*phi17, for n >= 0, with phi17 = A222132 = (1 + sqrt(17))/2, A052923(-n) = -(-2*i)^(-n)*S(-(n+2), i/2) = (i/2)^n*S(n, i/2), with i = sqrt(-1), and A006131(-(n+1)) = A052923(-n+1)/4 = -(i/2)^(n+1)*S(n-1, i/2), with the S-Chebyshev polynomials (see A049310), and S(-n, x) = -S(n-2, x), for n >= 1. - Wolfdieter Lang, Jan 04 2024

			c = 0.39038820320220756872767623199675962814339990317170255429982919663...

Index entries for sequences related to Chebyshev polynomials.

Cf. A006131, A010473, A049310, A052923, A174930, A189038, A222132.

RealDigits[(Sqrt[17] - 1)/8, 10, 120][[1]] (* Amiram Eldar, Jan 20 2023 *)
RealDigits[Root[4x^2+x-1,2],10,120][[1]] (* Harvey P. Dale, Jan 15 2024 *)

c = (-1 + sqrt(17))/8 = A189038 - 5/4 = A174930 - 5/8.

c = 1/phi17 = (-1 + phi17)/4, with phi17 = A222132. - Wolfdieter Lang, Jan 05 2024

A380897 Decimal expansion of (108)^(1/5).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Feb 07 2025

The effective degree of generic random walk on the barred-square graph (see Burda et al.).

			2.550849001251581665733095700385998546589800167...

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.

Cf. A222132.

Mathematica
```
RealDigits[(108)^(1/5),10,100][[1]]
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A365814 Decimal expansion of the largest root of the polynomial x^3 - 2*x^2 - 2*x + 2.

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A380895 Decimal expansion of (sqrt(17) + 1)/(4*sqrt(17)).

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A380896 Decimal expansion of (sqrt(17) - 1)/(4*sqrt(17)).

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A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

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A358945 Decimal expansion of the positive root of 4*x^2 + x - 1.

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A380897 Decimal expansion of (108)^(1/5).

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A365814 Decimal expansion of the largest root of the polynomial x^3 - 2x^2 - 2x + 2.