A010477 -id:A010477 - cp's OEIS frontend

A248249 Egyptian fraction representation of sqrt(21) (A010477) using a greedy function.

4, 2, 13, 177, 344766, 1649432522483, 3009384963228815398356405, 9085726642856091334926418336934724393317743509110, 200625769243543756748406312378876010708020812606355642597458369416042779347013395136132184521789202

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 21]]

A176014 Decimal expansion of (3+sqrt(21))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(21))/6 is A010684.

Also greatest eigenvalue of the 6 X 6 matrix [[3 0 0 3 0 0][0 0 0 0 1 0][0 3 0 0 3 0][0 0 0 0 1 0][0 0 3 0 0 3][0 0 0 0 1 0]]/3. It is conjectured that this is lim_{k->infinity} A005186(k+1)/A005186(k), i.e., the asymptotic growth rate of the number of numbers with the same total stopping time in the Collatz iteration. - Hugo Pfoertner, Sep 28 2020

			(3+sqrt(21))/6 = 1.26376261582597333443...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Ratio of successive terms in A005186, illustration of deviation from (3+sqrt(21))/6.
Index entries for algebraic numbers, degree 2.

Cf. A010477 (decimal expansion of sqrt(21)).
Cf. A010684 (repeat 1, 3), A136210, A136211.
Cf. A005186, A006577, A127824, A176866.

RealDigits[(3+Sqrt[21])/6,10,120][[1]] (* Harvey P. Dale, Jul 21 2023 *)

vecmax(mateigen([1,0,0,1,0,0; 0,0,0,0,1/3,0; 0,1,0,0,1,0; 0,0,0,0,1/3,0; 0,0,1,0,0,1; 0,0,0,0,1/3,0],1)[1]) \\ Hugo Pfoertner, Sep 28 2020

A176441 Decimal expansion of sqrt(210).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of sqrt(210) is A040195.

			sqrt(210) = 14.49137674618943857371...

Index entries for algebraic numbers, degree 2.

Cf. A010467 (decimal expansion of sqrt(10)), A010477 (decimal expansion of sqrt(21)), A176440 (decimal expansion of (14+sqrt(210))/4), A040195 (14 followed by (repeat 2, 28)).

RealDigits[Sqrt[210],10,120][[1]]  (* Harvey P. Dale, Apr 21 2011 *)

A010125 Continued fraction for sqrt(21).

Original entry on oeis.org

4, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1

Offset: 0

N. J. A. Sloane

			4.582575694955840006588047193... = 4 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 03 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A041032/A041033 (convergents), A010477 (decimal expansion).

ContinuedFraction[Sqrt[21],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1,1,2,1,1}] (* Harvey P. Dale, May 16 2020 *)

{ allocatemem(932245000); default(realprecision, 14000); x=contfrac(sqrt(21)); for (n=0, 20000, write("b010125.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

a(n) = gcd(Fibonacci(n), Fibonacci(n-6)), n > 0. - Gary Detlefs, Dec 29 2010

G.f.: (4 + x + x^2 + 2*x^3 + x^4 + x^5 + 4*x^6)/(1 - x^6). - Stefano Spezia, Jul 26 2025

A171537 Decimal expansion of sqrt(3/7).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -2 -1/2 | 7/2 -5/2>.

			sqrt(3/7) = 0.6546536707079771437982924562...

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[N[Sqrt[3/7],300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

equals A002194/A010465 = 3/A010477.

A176400 Decimal expansion of sqrt(483).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 17 2010

Continued fraction expansion of sqrt(483) is A040461.

			sqrt(483) = 21.97726097583591056720...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000

Cf. A010477 (decimal expansion of sqrt(21)), A010479 (decimal expansion of sqrt(23)), A176399 (decimal expansion of (21+sqrt(483))/7), A040461.

RealDigits[Sqrt[483],10,120][[1]] (* Harvey P. Dale, Oct 09 2015 *)

A041032 Numerators of continued fraction convergents to sqrt(21).

Original entry on oeis.org

4, 5, 9, 23, 32, 55, 472, 527, 999, 2525, 3524, 6049, 51916, 57965, 109881, 277727, 387608, 665335, 5710288, 6375623, 12085911, 30547445, 42633356, 73180801, 628079764, 701260565, 1329340329, 3359941223

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,110,0,0,0,0,0,-1).

Cf. A010477, A041033.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[21],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[21],50]] (* Harvey P. Dale, Apr 20 2012 *)

a(n) = 110*a(n-6)-a(n-12). G.f.: -(x^11 -4*x^10 +5*x^9 -9*x^8 +23*x^7 -32*x^6 -55*x^5 -32*x^4 -23*x^3 -9*x^2 -5*x -4)/((x^4 -5*x^2 +1) *(x^8 +5*x^6 +24*x^4 +5*x^2 +1)). - Colin Barker, Jul 16 2012

A041033 Denominators of continued fraction convergents to sqrt(21).

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 103, 115, 218, 551, 769, 1320, 11329, 12649, 23978, 60605, 84583, 145188, 1246087, 1391275, 2637362, 6665999, 9303361, 15969360, 137058241, 153027601, 290085842, 733199285, 1023285127

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,110,0,0,0,0,0,-1).

Cf. A010477, A041032.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[21],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Denominator[Convergents[Sqrt[21],30]] (* Harvey P. Dale, Apr 22 2013 *)
CoefficientList[Series[- (x^4 - x^3 + 2 x^2 + x + 1) (x^6 - 6 x^3 - 1)/((x^4 - 5 x^2 + 1) (x^8 + 5 x^6 + 24 x^4 + 5 x^2 + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)

a(n) = 110*a(n-6)-a(n-12). G.f.: -(x^4-x^3+2*x^2+x+1)*(x^6-6*x^3-1)/((x^4-5*x^2+1)*(x^8+5*x^6+24*x^4+5*x^2+1)). [Colin Barker, Jul 16 2012]

A365824 a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.

Original entry on oeis.org

1, 0, 5, 5, 30, 55, 205, 480, 1505, 3905, 11430, 30955, 88105, 242880, 683405, 1897805, 5314830, 14803855, 41378005, 115397280, 322287305, 899273705, 2510710230, 7007078755, 19560629905, 54596023680, 152399173205, 425379291605, 1187375157630, 3314271615655

Offset: 0

Wolfdieter Lang, Nov 20 2023

This sequence {a(n)} appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with b(n) = A015440(n-1), with A015440(-1) = 0, as phi21^n = a(n) + b(n)*phi21(n), for n >= 0. But the later given formulas in terms of scaled Chebyshev polynomials, called here {S21(n)}, are valid also for negative n values, i.e., for nonnegative powers of 1/phi21 = (-1 + sqrt(21))/10 = 0.35825756949... = A367453.

Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(21))/2 = A222134 = 2.791287...

			phi21^2 = a(2) + b(2)*phi(n) = 5 + phi21 = 7.79128784..., a real algebraic integer in Q(sqrt(21)).
(1/phi21)^2 = a(-2) + b(-2)*phi21 = (1/25)*(6 - phi21) = 0.12834848..., a real algebraic number in Q(sqrt(21)).

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A010477 (sqrt(21)), A015440, A049310, A222134, A367453.

LinearRecurrence[{1,5},{1,0},50] (* Paolo Xausa, Nov 21 2023 *)

a(n) = abs([1, 3; 1, -2]^(n-2)*[5; 5])[2, 1] \\ Thomas Scheuerle, Nov 20 2023

a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.
G.f.: (1 - x)/(1 - x - 5*x^2).
a(n) = S21(n+1) - S21(n), for n >= 0, where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
The above mentioned sequence {b(n)} has terms b(n) = A015440(n-1) =  S21(n), for n >= 0, with the same recurrence as {a(n)} but with b(0) = 0 and b(1) = 1, and g.f. x/(1 - x - 5*x^2).
The formula for negative indices of S is: S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.

A017967 Powers of sqrt(21) rounded down.

Original entry on oeis.org

1, 4, 21, 96, 441, 2020, 9261, 42439, 194481, 891223, 4084101, 18715701, 85766121, 393029741, 1801088541, 8253624572, 37822859361, 173326116021, 794280046581, 3639848436450, 16679880978201, 76436817165460

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A010477 (sqrt(21)).

[Floor(Sqrt(21^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Floor[(Sqrt[21])^Range[0,30]] (* Harvey P. Dale, May 08 2018 *)

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

a(n) = floor(sqrt(21^n)). - Vincenzo Librandi, Jun 24 2011

cp's OEIS Frontend

A248249 Egyptian fraction representation of sqrt(21) (A010477) using a greedy function.

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A176014 Decimal expansion of (3+sqrt(21))/6.

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A176441 Decimal expansion of sqrt(210).

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A010125 Continued fraction for sqrt(21).

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A171537 Decimal expansion of sqrt(3/7).

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A176400 Decimal expansion of sqrt(483).

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A041032 Numerators of continued fraction convergents to sqrt(21).

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A041033 Denominators of continued fraction convergents to sqrt(21).

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A365824 a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.