A010521 -id:A010521

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A248293

Egyptian fraction representation of sqrt(69) (A010521) using a greedy function.

+20
0

8, 4, 18, 937, 933269, 1035335826584, 1922586201513701668252744, 28276178347455966021225105018046994195665521584589, 833556138210674401337075496134582593689166273775276908669899884379507156146934822563063380503158977

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

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..8.

MATHEMATICA

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[ 69]]

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Oct 04 2014

STATUS

approved

A010148

Continued fraction for sqrt(69).

+10
4

8, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1

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

OFFSET

0,1

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

A. J. van der Poorten, An introduction to continued fractions, Unpublished.

A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]

G. Xiao, Contfrac

Index entries for continued fractions for constants

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

EXAMPLE

8.306623862918074852584262744... = 8 + 1/(3 + 1/(3 + 1/(1 + 1/(4 + ...)))). - Harry J. Smith, Jun 08 2009

MATHEMATICA

ContinuedFraction[Sqrt[69], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)

PadRight[{8}, 120, {16, 3, 3, 1, 4, 1, 3, 3}] (* Harvey P. Dale, Jan 25 2024 *)

PROG

(PARI) { allocatemem(932245000); default(realprecision, 22000); x=contfrac(sqrt(69)); for (n=0, 20000, write("b010148.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

CROSSREFS

Cf. A010521 Decimal expansion. - Harry J. Smith, Jun 08 2009

KEYWORD

nonn,cofr

AUTHOR

N. J. A. Sloane

STATUS

approved

A041120

Numerators of continued fraction convergents to sqrt(69).

+10
2

8, 25, 83, 108, 515, 623, 2384, 7775, 126784, 388127, 1291165, 1679292, 8008333, 9687625, 37071208, 120901249, 1971491192, 6035374825, 20077615667, 26112990492, 124529577635, 150642568127, 576457282016, 1880014414175, 30656687908816, 93850078140623

(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 (0,0,0,0,0,0,0,15550,0,0,0,0,0,0,0,-1).

FORMULA

G.f.: -(x^15 -8*x^14 +25*x^13 -83*x^12 +108*x^11 -515*x^10 +623*x^9 -2384*x^8 -7775*x^7 -2384*x^6 -623*x^5 -515*x^4 -108*x^3 -83*x^2 -25*x -8) / (x^16 -15550*x^8 +1). - Colin Barker, Nov 10 2013

MATHEMATICA

Numerator[Convergents[Sqrt[69], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

CROSSREFS

Cf. A010521, A041121.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 10 2013

STATUS

approved

A041121

Denominators of continued fraction convergents to sqrt(69).

+10
2

1, 3, 10, 13, 62, 75, 287, 936, 15263, 46725, 155438, 202163, 964090, 1166253, 4462849, 14554800, 237339649, 726573747, 2417060890, 3143634637, 14991599438, 18135234075, 69397301663, 226327139064, 3690631526687, 11298221719125, 37585296684062, 48883518403187

(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 (0,0,0,0,0,0,0,15550,0,0,0,0,0,0,0,-1).

FORMULA

G.f.: -(x^14 -3*x^13 +10*x^12 -13*x^11 +62*x^10 -75*x^9 +287*x^8 -936*x^7 -287*x^6 -75*x^5 -62*x^4 -13*x^3 -10*x^2 -3*x -1) / (x^16 -15550*x^8 +1). - Colin Barker, Nov 13 2013

a(n) = 15550*a(n-8) - a(n-16). - Vincenzo Librandi, Dec 11 2013

MATHEMATICA

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[69], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)

Denominator[Convergents[Sqrt[69], 30]] (* Vincenzo Librandi, Dec 11 2013 *)

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 15550, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 3, 10, 13, 62, 75, 287, 936, 15263, 46725, 155438, 202163, 964090, 1166253, 4462849, 14554800}, 30] (* Harvey P. Dale, Oct 18 2015 *)

PROG

(Magma) I:=[1, 3, 10, 13, 62, 75, 287, 936, 15263, 46725, 155438, 202163, 964090, 1166253, 4462849, 14554800]; [n le 16 select I[n] else 15550*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

CROSSREFS

Cf. A041120, A010148, A020826, A010521.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 13 2013

STATUS

approved

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