A177935 -id:A177935

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A010173

Continued fraction for sqrt(107).

+10
4

10, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1, 9, 1, 2, 20, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..999

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).

FORMULA

G.f.: (-10*x^6 - 2*x^5 - x^4 - 9*x^3 - x^2 - 2*x - 10)/(x^6 - 1). - Chai Wah Wu, Oct 02 2021

MATHEMATICA

ContinuedFraction[Sqrt[107], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2011 *)

PROG

(Python)

from sympy import sqrt

from sympy.ntheory.continued_fraction import continued_fraction_iterator

def aupton(terms):

gen = continued_fraction_iterator(sqrt(107))

return [next(gen) for i in range(terms)]

print(aupton(82)) # Michael S. Branicky, Oct 02 2021

CROSSREFS

Cf. A177935 (decimal expansion), A041192/A041193 (convergents).

KEYWORD

nonn,cofr,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A177934

Decimal expansion of sqrt(71216963807).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

6,1

COMMENTS

Continued fraction expansion of sqrt(71216963807) is 266865 followed by (repeat 15, 533730).

sqrt(71216963807) = sqrt(11)*sqrt(19)*sqrt(107)*sqrt(179)*sqrt(17791).

LINKS

Table of n, a(n) for n=6..110.

EXAMPLE

sqrt(71216963807) = 266865.06666665833952872396...

MATHEMATICA

RealDigits[Sqrt[71216963807], 10, 120][[1]] (* Harvey P. Dale, Jul 31 2021 *)

CROSSREFS

Cf. A010468 (decimal expansion of sqrt(11)), A010475 (decimal expansion of sqrt(19)), A177935 (decimal expansion of sqrt(107)), A177936 (decimal expansion of sqrt(179)), A177937 (decimal expansion of sqrt(17791)), A177933 (decimal expansion of (232405+sqrt(71216963807))/348378).

KEYWORD

cons,nonn

AUTHOR

Klaus Brockhaus, May 15 2010

STATUS

approved

A041192

Numerators of continued fraction convergents to sqrt(107).

+10
3

10, 21, 31, 300, 331, 962, 19571, 40104, 59675, 577179, 636854, 1850887, 37654594, 77160075, 114814669, 1110492096, 1225306765, 3561105626, 72447419285, 148455944196, 220903363481, 2136586215525, 2357489579006

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1924,0,0,0,0,0,-1).

FORMULA

a(n) = 1924*a(n-6)-a(n-12). G.f.: -(x^11-10*x^10+21*x^9-31*x^8+300*x^7-331*x^6-962*x^5-331*x^4-300*x^3-31*x^2-21*x-10)/(x^12-1924*x^6+1). [Colin Barker, Jul 19 2012]

MATHEMATICA

Numerator[Convergents[Sqrt[107], 30]] (* Vincenzo Librandi, Oct 26 2013 *)

CROSSREFS

Cf. A041193 (denominators), A010173 (continued fraction), A177935 (decimal expansion).

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A041193

Denominators of continued fraction convergents to sqrt(107).

+10
3

1, 2, 3, 29, 32, 93, 1892, 3877, 5769, 55798, 61567, 178932, 3640207, 7459346, 11099553, 107355323, 118454876, 344265075, 7003756376, 14351777827, 21355534203, 206551585654, 227907119857, 662365825368

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1924,0,0,0,0,0,-1).

FORMULA

a(n) = 1924*a(n-6)-a(n-12). G.f.: -(x^10-2*x^9+3*x^8-29*x^7+32*x^6 -93*x^5 -32*x^4-29*x^3-3*x^2-2*x-1) / (x^12-1924*x^6+1). - Colin Barker, Jul 19 2012

MATHEMATICA

Denominator[Convergents[Sqrt[107], 30]] (* or *) LinearRecurrence[ {0, 0, 0, 0, 0, 1924, 0, 0, 0, 0, 0, -1}, {1, 2, 3, 29, 32, 93, 1892, 3877, 5769, 55798, 61567, 178932}, 30] (* Harvey P. Dale, Aug 30 2013 *)

CoefficientList[Series[- (x^10 - 2 x^9 + 3 x^8 - 29 x^7 + 32 x^6 - 93 x^5 - 32 x^4 - 29 x^3 - 3 x^2 - 2 x - 1)/(x^12 - 1924 x^6 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 24 2013 *)

CROSSREFS

Cf. A041192 (numerators), A010173 (continued fraction), A177935 (decimal expansion).

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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