A010544 -id:A010544 - cp's OEIS frontend

A248316 Egyptian fraction representation of sqrt(93) (A010544) using a greedy function.

9, 2, 7, 1261, 1675816, 88347178296709, 9125661908898813209593366366, 121027088765475470246692180437471007789578710116057334417, 12633191396046337132562053250878765047674173616082104244017001032417219816643782643353462090455648430033589876568764

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[ 93]]

A010164 Continued fraction for sqrt(93).

Original entry on oeis.org

9, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1

Offset: 0

N. J. A. Sloane

			9.643650760992954995760031047... = 9 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))). - _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, 0, 0, 1).

Cf. A010544 Decimal expansion. - Harry J. Smith, Jun 11 2009

ContinuedFraction[Sqrt[93],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)
PadRight[{9},120,{18,1,1,1,4,6,4,1,1,1}] (* Harvey P. Dale, Mar 30 2019 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(93)); for (n=0, 20000, write("b010164.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 11 2009

A041166 Numerators of continued fraction convergents to sqrt(93).

Original entry on oeis.org

9, 10, 19, 29, 135, 839, 3491, 4330, 7821, 12151, 226539, 238690, 465229, 703919, 3280905, 20389349, 84838301, 105227650, 190065951, 295293601, 5505350769, 5800644370, 11305995139, 17106639509, 79732553175, 495501958559, 2061740387411, 2557242345970

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 (0,0,0,0,0,0,0,0,0,24302,0,0,0,0,0,0,0,0,0,-1).

Cf. A010544, A041167.

Numerator[Convergents[Sqrt[93], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,24302,0,0,0,0,0,0,0,0,0,-1},{9,10,19,29,135,839,3491,4330,7821,12151,226539,238690,465229,703919,3280905,20389349,84838301,105227650,190065951,295293601},30] (* Harvey P. Dale, Apr 10 2023 *)

G.f.: -(x^19 -9*x^18 +10*x^17 -19*x^16 +29*x^15 -135*x^14 +839*x^13 -3491*x^12 +4330*x^11 -7821*x^10 -12151*x^9 -7821*x^8 -4330*x^7 -3491*x^6 -839*x^5 -135*x^4 -29*x^3 -19*x^2 -10*x -9) / (x^20 -24302*x^10 +1). - Colin Barker, Nov 10 2013

More terms from Colin Barker, Nov 10 2013

A041167 Denominators of continued fraction convergents to sqrt(93).

Original entry on oeis.org

1, 1, 2, 3, 14, 87, 362, 449, 811, 1260, 23491, 24751, 48242, 72993, 340214, 2114277, 8797322, 10911599, 19708921, 30620520, 570878281, 601498801, 1172377082, 1773875883, 8267880614, 51381159567, 213792518882, 265173678449, 478966197331, 744139875780

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 (0,0,0,0,0,0,0,0,0,24302,0,0,0,0,0,0,0,0,0,-1).

Cf. A041166, A010164, A020850, A010544.

I:=[1, 1, 2, 3, 14, 87, 362, 449, 811, 1260, 23491, 24751, 48242, 72993, 340214, 2114277, 8797322, 10911599, 19708921, 30620520]; [n le 20 select I[n] else 24302*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[93], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^18 -x^17 +2*x^16 -3*x^15 +14*x^14 -87*x^13 +362*x^12 -449*x^11 +811*x^10 -1260*x^9 -811*x^8 -449*x^7 -362*x^6 -87*x^5 -14*x^4 -3*x^3 -2*x^2 -x -1) / (x^20 -24302*x^10 +1). - Colin Barker, Nov 14 2013

a(n) = 24302*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 12 2013

More terms from Colin Barker, Nov 14 2013

A176108 Decimal expansion of (9+sqrt(93))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (9+sqrt(93))/6 is A010707.

			(9+sqrt(93))/6 = 3.10727512683215916596...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010544 (decimal expansion of sqrt(93)), A010707 (repeat 3, 9).

RealDigits[(9+Sqrt[93])/6,10,120][[1]] (* Harvey P. Dale, Nov 12 2022 *)

A011084 Decimal expansion of 4th root of 93.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.105422799071481505658083732285614...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A010544.

sqrtn(93,4) \\ Charles R Greathouse IV, Apr 16 2014

Equals sqrt(A010544). - Michel Marcus, Aug 13 2023

A176516 Decimal expansion of (9+sqrt(93))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (9+sqrt(93))/2 is A010707 preceded by 9.

			(9+sqrt(93))/2 = 9.32182538049647749788...

Cf. A010544 (decimal expansion of sqrt(93)), A010707 (repeat 3, 9).

RealDigits[(9+Sqrt[93])/2,10,120][[1]] (* Harvey P. Dale, Aug 07 2025 *)

A177003 Decimal expansion of (7+sqrt(93))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2010

Continued fraction expansion of (7+sqrt(93))/6 is A157810.

			(7+sqrt(93))/6 = 2.77394179349882583262...

Cf. A010544 (decimal expansion of sqrt(93)), A157810 (repeat 2, 1, 3, 2).

RealDigits[(7+Sqrt[93])/6,10,120][[1]] (* Harvey P. Dale, Aug 23 2014 *)

(sqrt(93)+7)/6 \\ Charles R Greathouse IV, Apr 25 2016

cp's OEIS Frontend

A248316 Egyptian fraction representation of sqrt(93) (A010544) using a greedy function.

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A010164 Continued fraction for sqrt(93).

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A041166 Numerators of continued fraction convergents to sqrt(93).

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A041167 Denominators of continued fraction convergents to sqrt(93).

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A176108 Decimal expansion of (9+sqrt(93))/6.

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A011084 Decimal expansion of 4th root of 93.

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A176516 Decimal expansion of (9+sqrt(93))/2.

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A177003 Decimal expansion of (7+sqrt(93))/6.

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