A222135 -id:A222135 - cp's OEIS frontend

A222134 Decimal expansion of sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))).

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

Offset: 1

Jaroslav Krizek, Feb 08 2013

c^n = A015440(n) + A015440(n-1) * A222135, where c = (1 + sqrt(21))/2 and A222135 = (-1 + sqrt(21))/2. - Gary W. Adamson, Nov 27 2023

			2.791287847477920003294023596864...

Index entries for algebraic numbers, degree 2.

Cf. A090458, A107905, A222135, A015440.

Mathematica
```
RealDigits[(1 + Sqrt[21])/2, 10, 130]
```

(sqrt(21) + 1)/2 \\ Charles R Greathouse IV, Feb 07 2025

Equals (sqrt(21) + 1)/2 = A090458 - 1 = A107905 - 2 = A222135 + 1.
sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))) - 1 = sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))) = A222135.
Minimal polynomial: x^2 - x - 5. - Stefano Spezia, Jul 02 2025

A367453 Decimal expansion of (-1 + sqrt(21))/10 = 1/A222134.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 20 2023

Positive root of the minimal polynomial x^2 + 1/5 - 1/5. The negative root is -(1/5)*A222134 = -0.558257569...
c^n = A(-n) + B(-n)*phi21, and A(n) = S21(n+1) - S21(n) = A365824(n), with phi21 = A222134, and B(n) = S21(n) = A015440(n-1), where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
The formula for negative indices of S is S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.

			c = 0.3582575694955840006588047193728008488984456...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A010477, A049310, A222134, A222135, A365824.

First[RealDigits[(Sqrt[21]-1)/10,10,100]] (* Paolo Xausa, Nov 21 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^(n-1)*(quadgen(84)-1))) \\ Michal Paulovic, Nov 20 2023

c = 1/phi21 = (1/5)*(1 - phi21), with phi21 = (1 + sqrt(21))/2 = A222134, hence an algebraic number of the real quadratic number field Q(sqrt(21)) but not an algebraic integer like phi24.
Equals (A010477-1)/10. - R. J. Mathar, Nov 21 2023
Equals 2*A222135/10. - Hugo Pfoertner, Mar 21 2024

cp's OEIS Frontend

A222134 Decimal expansion of sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))).

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