A010523 -id:A010523 - cp's OEIS frontend

A248295 Egyptian fraction representation of sqrt(71) (A010523) using a greedy function.

8, 3, 11, 525, 386544, 639498711870, 1018235602235689213572994, 6335607869766803762689695208858285361004070429148, 42457213694266417320054923496312615766199040305766336893524891089914272708684998227290613582884885

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[ 71]]

A010150 Continued fraction for sqrt(71).

Original entry on oeis.org

8, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1

Offset: 0

N. J. A. Sloane

			8.426149773176358630634139906... = 8 + 1/(2 + 1/(2 + 1/(1 + 1/(7 + ...)))). - _Harry J. Smith_, Jun 08 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. A010523 Decimal expansion. - Harry J. Smith, Jun 08 2009

ContinuedFraction[Sqrt[71],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,2,2,1,7,1,2,2}] (* Harvey P. Dale, Apr 17 2015 *)

{ allocatemem(932245000); default(realprecision, 20000); x=contfrac(sqrt(71)); for (n=0, 20000, write("b010150.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

A041124 Numerators of continued fraction convergents to sqrt(71).

Original entry on oeis.org

8, 17, 42, 59, 455, 514, 1483, 3480, 57163, 117806, 292775, 410581, 3166842, 3577423, 10321688, 24220799, 397854472, 819929743, 2037713958, 2857643701, 22041219865, 24898863566, 71838946997, 168576757560, 2769067067957, 5706710893474, 14182488854905

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 (0,0,0,0,0,0,0,6960,0,0,0,0,0,0,0,-1).

Cf. A010523, A041125.

Numerator[Convergents[Sqrt[71], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,6960,0,0,0,0,0,0,0,-1},{8,17,42,59,455,514,1483,3480,57163,117806,292775,410581,3166842,3577423,10321688,24220799},40] (* Harvey P. Dale, Aug 14 2023 *)

G.f.: -(x^15 -8*x^14 +17*x^13 -42*x^12 +59*x^11 -455*x^10 +514*x^9 -1483*x^8 -3480*x^7 -1483*x^6 -514*x^5 -455*x^4 -59*x^3 -42*x^2 -17*x -8) / (x^16 -6960*x^8 +1). - Colin Barker, Nov 05 2013

More terms from Colin Barker, Nov 05 2013

A041125 Denominators of continued fraction convergents to sqrt(71).

Original entry on oeis.org

1, 2, 5, 7, 54, 61, 176, 413, 6784, 13981, 34746, 48727, 375835, 424562, 1224959, 2874480, 47216639, 97307758, 241832155, 339139913, 2615811546, 2954951459, 8525714464, 20006380387, 328627800656, 677261981699, 1683151764054, 2360413745753, 18206047984325

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 (0,0,0,0,0,0,0,6960,0,0,0,0,0,0,0,-1).

Cf. A041124, A010150, A020828, A010523.

I:=[1, 2, 5, 7, 54, 61, 176, 413, 6784, 13981, 34746, 48727, 375835, 424562, 1224959, 2874480]; [n le 16 select I[n] else 6960*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[71],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Denominator[Convergents[Sqrt[71], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,6960,0,0,0,0,0,0,0,-1},{1,2,5,7,54,61,176,413,6784,13981,34746,48727,375835,424562,1224959,2874480},30] (* Harvey P. Dale, Apr 09 2022 *)

G.f.: -(x^14 -2*x^13 +5*x^12 -7*x^11 +54*x^10 -61*x^9 +176*x^8 -413*x^7 -176*x^6 -61*x^5 -54*x^4 -7*x^3 -5*x^2 -2*x -1) / (x^16 -6960*x^8 +1). - Colin Barker, Nov 13 2013

a(n) = 6960*a(n-8) - a(n-16). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 13 2013

A176714 Decimal expansion of sqrt(16394397).

Original entry on oeis.org

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

Offset: 4

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of sqrt(16394397) is 4048 followed by (repeat 1, 2023, 2, 2023, 1, 8096).

sqrt(16394397) = sqrt(3)*sqrt(19)*sqrt(71)*sqrt(4051).

			sqrt(16394397) = 4048.99950605084663055493...

Cf. A002194 (decimal expansion of sqrt(3)), A010475 (decimal expansion of sqrt(19)), A010523 (decimal expansion of sqrt(71)), A176715 (decimal expansion of sqrt(4051)), A176713 (decimal expansion of (4047+sqrt(16394397))/142).

Digits:=100: evalf(sqrt(16394397)); # Wesley Ivan Hurt, Jul 07 2014

RealDigits[Sqrt[16394397], 10, 100, 3] (* Wesley Ivan Hurt, Jul 07 2014 *)

cp's OEIS Frontend

A248295 Egyptian fraction representation of sqrt(71) (A010523) using a greedy function.

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A010150 Continued fraction for sqrt(71).

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A041124 Numerators of continued fraction convergents to sqrt(71).

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A041125 Denominators of continued fraction convergents to sqrt(71).

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Magma

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A176714 Decimal expansion of sqrt(16394397).

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