A010477 -id:A010477 - cp's OEIS frontend

A020841 Decimal expansion of 1/sqrt(84).

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

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A010477.

RealDigits[1/Sqrt[84],10,120][[1]] (* Harvey P. Dale, Nov 27 2013 *)

Equals 1/(2*A010477). - Stefano Spezia, Jan 19 2025

A176106 Decimal expansion of (21+5*sqrt(21))/14.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (21+5*sqrt(21))/14 is A010705.

			(21+5*sqrt(21))/14 = 3.13663417676994285949...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010477 (decimal expansion of sqrt(21)), A010705 (repeat 3, 7).

RealDigits[(21+5*Sqrt[21])/14,10,120][[1]] (* Harvey P. Dale, Apr 27 2017 *)

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

A017968 Powers of sqrt(21) rounded to nearest integer.

Original entry on oeis.org

1, 5, 21, 96, 441, 2021, 9261, 42439, 194481, 891224, 4084101, 18715702, 85766121, 393029742, 1801088541, 8253624572, 37822859361, 173326116021, 794280046581, 3639848436450, 16679880978201, 76436817165460, 350277500542221, 1605173160474663, 7355827511386641

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010477 (sqrt(21)), A017967.

Bisection gives A009965 (even part).

[Round(Sqrt(21)^n): n in [0..30]]; // Vincenzo Librandi, Nov 20 2011

a:= n-> round(sqrt(21)^n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2022

Floor[(Sqrt[21])^Range[0,25]+1/2] (* Harvey P. Dale, Sep 22 2011 *)

a(n)=round(sqrt(21)^n) \\ Charles R Greathouse IV, Nov 18 2011

from math import isqrt
def A017968(n): return (m:=isqrt(k:=21**n))+int((k-m*(m+1)<<2)>=1) # Chai Wah Wu, Jul 29 2022

A176435 Decimal expansion of (21+5*sqrt(21))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (21+5*sqrt(21))/6 is A010705 preceded by 7.

			(21+5*sqrt(21))/6 = 7.31881307912986667215...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010477 (decimal expansion of sqrt(21)), A010705 (repeat 3, 7).

RealDigits[(21+5*Sqrt[21])/6,10,120][[1]] (* Harvey P. Dale, Feb 20 2022 *)

A178131 Decimal expansion of (11+3*sqrt(21))/17.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 20 2010

Continued fraction expansion of (11+3*sqrt(21))/17 is A131800.

			(11+3*sqrt(21))/17 = 1.45574865205103058939...

Cf. A010477 (decimal expansion of sqrt(21)), A131800 (repeat 1, 2, 5, 6).

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A020841 Decimal expansion of 1/sqrt(84).

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A176106 Decimal expansion of (21+5*sqrt(21))/14.

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

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A017968 Powers of sqrt(21) rounded to nearest integer.

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A176435 Decimal expansion of (21+5*sqrt(21))/6.

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A178131 Decimal expansion of (11+3*sqrt(21))/17.

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