A010548 -id:A010548 - cp's OEIS frontend

A248320 Egyptian fraction representation of sqrt(97) (A010548) using a greedy function.

9, 2, 3, 65, 7151, 80022924, 30599691861141025, 941764378394111110366748005603120790, 1207204053149486733914427645845997404101266838687995330054616662472844167

Offset: 0

Robert G. Wilson v, Oct 05 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[ 97]]

A010168 Continued fraction for sqrt(97).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			9.848857801796104721746211414... = 9 + 1/(1 + 1/(5 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 12 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 (0,0,0,0,0,0,0,0,0,0,1).

Cf. A010548 (decimal expansion).

Cf. A041174/A041175 (convergents).

ContinuedFraction[Sqrt[97],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)
PadRight[{9},120,{18,1,5,1,1,1,1,1,1,5,1}] (* Harvey P. Dale, Jun 07 2016 *)

{ allocatemem(932245000); default(realprecision, 15000); x=contfrac(sqrt(97)); for (n=0, 20000, write("b010168.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 12 2009

A041174 Numerators of continued fraction convergents to sqrt(97).

Original entry on oeis.org

9, 10, 59, 69, 128, 197, 325, 522, 847, 4757, 5604, 105629, 111233, 661794, 773027, 1434821, 2207848, 3642669, 5850517, 9493186, 53316447, 62809633, 1183889841, 1246699474, 7417387211, 8664086685, 16081473896, 24745560581, 40827034477, 65572595058

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 (0,0,0,0,0,0,0,0,0,0,11208,0,0,0,0,0,0,0,0,0,0,1).

Cf. A010548, A041175, A010168, A020854.

Numerator[Convergents[Sqrt[97], 30]] (* Vincenzo Librandi, Oct 30 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,11208,0,0,0,0,0,0,0,0,0,0,1},{9,10,59,69,128,197,325,522,847,4757,5604,105629,111233,661794,773027,1434821,2207848,3642669,5850517,9493186,53316447,62809633},30] (* Harvey P. Dale, Aug 02 2021 *)

G.f.: -(x^21 -9*x^20 +10*x^19 -59*x^18 +69*x^17 -128*x^16 +197*x^15 -325*x^14 +522*x^13 -847*x^12 +4757*x^11 +5604*x^10 +4757*x^9 +847*x^8 +522*x^7 +325*x^6 +197*x^5 +128*x^4 +69*x^3 +59*x^2 +10*x +9) / (x^22 +11208*x^11 -1). - Colin Barker, Nov 14 2013

More terms from Colin Barker, Nov 14 2013

A041175 Denominators of continued fraction convergents to sqrt(97).

Original entry on oeis.org

1, 1, 6, 7, 13, 20, 33, 53, 86, 483, 569, 10725, 11294, 67195, 78489, 145684, 224173, 369857, 594030, 963887, 5413465, 6377352, 120205801, 126583153, 753121566, 879704719, 1632826285, 2512531004, 4145357289, 6657888293, 10803245582, 60674116203, 71477361785

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 (0,0,0,0,0,0,0,0,0,0,11208,0,0,0,0,0,0,0,0,0,0,1).

Cf. A041174, A010168, A020854, A010548.

I:=[1, 1, 6, 7, 13, 20, 33, 53, 86, 483, 569, 10725, 11294, 67195, 78489, 145684, 224173, 369857, 594030, 963887, 5413465, 6377352]; [n le 22 select I[n] else 11208*Self(n-11)+Self(n-22): n in [1..40]]; // Vincenzo Librandi, Dec 12 2013

Denominator[Convergents[Sqrt[97], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^20 -x^19 +6*x^18 -7*x^17 +13*x^16 -20*x^15 +33*x^14 -53*x^13 +86*x^12 -483*x^11 +569*x^10 +483*x^9 +86*x^8 +53*x^7 +33*x^6 +20*x^5 +13*x^4 +7*x^3 +6*x^2 +x +1) / (x^22 +11208*x^11 -1). - Colin Barker, Nov 14 2013

a(n) = 11208*a(n-11) + a(n-22). - Vincenzo Librandi, Dec 12 2013

More terms from Colin Barker, Nov 14 2013

A177159 Decimal expansion of sqrt(4171).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 03 2010

Continued fraction expansion of sqrt(4171) is 64 followed by (repeat 1, 1, 2, 1, 1, 128).

sqrt(4171) = sqrt(43)*sqrt(97).

			sqrt(4171) = 64.58327956987009497650...

Cf. A010497 (decimal expansion of sqrt(43)), A010548 (decimal expansion of sqrt(97)), A177158 (decimal expansion of (103+2*sqrt(4171))/162).

cp's OEIS Frontend

A248320 Egyptian fraction representation of sqrt(97) (A010548) using a greedy function.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A010168 Continued fraction for sqrt(97).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A041174 Numerators of continued fraction convergents to sqrt(97).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A041175 Denominators of continued fraction convergents to sqrt(97).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A177159 Decimal expansion of sqrt(4171).

Views

Author

Keywords

Comments

Examples

Crossrefs