A010521 -id:A010521 - cp's OEIS frontend

A248293 Egyptian fraction representation of sqrt(69) (A010521) using a greedy function.

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

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[ 69]]

A010148 Continued fraction for sqrt(69).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

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A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]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

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

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 *)

{ 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

A041120 Numerators of continued fraction convergents to sqrt(69).

Original entry on oeis.org

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

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

Cf. A010521, A041121.

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

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

More terms from Colin Barker, Nov 10 2013

A041121 Denominators of continued fraction convergents to sqrt(69).

Original entry on oeis.org

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

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

Cf. A041120, A010148, A020826, A010521.

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

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 *)

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

More terms from Colin Barker, Nov 13 2013

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A248293 Egyptian fraction representation of sqrt(69) (A010521) using a greedy function.

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A010148 Continued fraction for sqrt(69).

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A041120 Numerators of continued fraction convergents to sqrt(69).

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A041121 Denominators of continued fraction convergents to sqrt(69).

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