A010542 -id:A010542 - cp's OEIS frontend

A248314 Egyptian fraction representation of sqrt(91) (A010542) using a greedy function.

9, 2, 26, 1075, 4112692, 27635607215221, 849264815973068493208894568, 4841632147722920600382393247090367280521902842548984840, 266615765339465306160951850397944334901944400721855522651965673953142506552009472018442954194685982876108261727

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[ 91]]

A010162 Continued fraction for sqrt(91).

Original entry on oeis.org

9, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5

Offset: 0

N. J. A. Sloane

			9.539392014169456491526215860... = 9 + 1/(1 + 1/(1 + 1/(5 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 11 2009

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, 0, 0, 1).

Cf. A010542 Decimal expansion. - Harry J. Smith, Jun 11 2009

ContinuedFraction[Sqrt[91],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(91)); for (n=0, 20000, write("b010162.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 11 2009

A041162 Numerators of continued fraction convergents to sqrt(91).

Original entry on oeis.org

9, 10, 19, 105, 124, 725, 849, 1574, 29181, 30755, 59936, 330435, 390371, 2282290, 2672661, 4954951, 91861779, 96816730, 188678509, 1040209275, 1228887784, 7184648195, 8413535979, 15598184174, 289180851111, 304779035285, 593959886396, 3274578467265

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 (0,0,0,0,0,0,0,3148,0,0,0,0,0,0,0,-1).

Cf. A010542, A041163.

Numerator[Convergents[Sqrt[91], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^15 -9*x^14 +10*x^13 -19*x^12 +105*x^11 -124*x^10 +725*x^9 -849*x^8 -1574*x^7 -849*x^6 -725*x^5 -124*x^4 -105*x^3 -19*x^2 -10*x -9) / (x^16 -3148*x^8 +1). - Colin Barker, Nov 05 2013

More terms from Colin Barker, Nov 05 2013

A041163 Denominators of continued fraction convergents to sqrt(91).

Original entry on oeis.org

1, 1, 2, 11, 13, 76, 89, 165, 3059, 3224, 6283, 34639, 40922, 239249, 280171, 519420, 9629731, 10149151, 19778882, 109043561, 128822443, 753155776, 881978219, 1635133995, 30314390129, 31949524124, 62263914253, 343269095389, 405533009642, 2370934143599

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 (0,0,0,0,0,0,0,3148,0,0,0,0,0,0,0,-1).

Cf. A041162, A010162, A020848, A010542.

I:=[1, 1, 2, 11, 13, 76, 89, 165, 3059, 3224, 6283, 34639, 40922, 239249, 280171, 519420]; [n le 16 select I[n] else 3148*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[91], 30]] (* Bruno Berselli, Nov 14 2013 *)
CoefficientList[Series[-(x^14 - x^13 + 2 x^12 - 11 x^11 + 13 x^10 - 76 x^9 + 89 x^8 - 165 x^7 - 89 x^6 - 76 x^5 - 13 x^4 - 11 x^3 - 2 x^2 - x - 1)/(x^16 - 3148 x^8 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^14 -x^13 +2*x^12 -11*x^11 +13*x^10 -76*x^9 +89*x^8 -165*x^7 -89*x^6 -76*x^5 -13*x^4 -11*x^3 -2*x^2 -x -1) / (x^16 -3148*x^8 +1). - Colin Barker, Nov 14 2013

a(n) = 3148*a(n-8) - a(n-16). - Vincenzo Librandi, Dec 12 2013

More terms from Colin Barker, Nov 14 2013

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A248314 Egyptian fraction representation of sqrt(91) (A010542) using a greedy function.

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A010162 Continued fraction for sqrt(91).

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A041162 Numerators of continued fraction convergents to sqrt(91).

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A041163 Denominators of continued fraction convergents to sqrt(91).

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