A010537 -id:A010537 - cp's OEIS frontend

A248309 Egyptian fraction representation of sqrt(86) (A010537) using a greedy function.

9, 4, 43, 2758, 10003866, 210627752029830, 52347107682242256851283255963, 2888760102328257324292867347514624728389388637643413378835, 73966122168790045965795439815232439142274674774329779302853705553926323866554092485061114120398281226411275872002645

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[ 86]]

A010159 Continued fraction for sqrt(86).

Original entry on oeis.org

9, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1

Offset: 0

N. J. A. Sloane

			9.273618495495703752516416073... = 9 + 1/(3 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 10 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. A010537 (decimal expansion). - Harry J. Smith, Jun 10 2009

ContinuedFraction[Sqrt[86],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
PadRight[{9},120,{18,3,1,1,1,8,1,1,1,3}] (* Harvey P. Dale, Dec 13 2018 *)

from sympy import sqrt
from sympy.ntheory.continued_fraction import continued_fraction_iterator
def aupton(terms):
    gen = continued_fraction_iterator(sqrt(86))
    return [next(gen) for i in range(terms)]
print(aupton(78)) # Michael S. Branicky, Sep 03 2021

A041152 Numerators of continued fraction convergents to sqrt(86).

Original entry on oeis.org

9, 28, 37, 65, 102, 881, 983, 1864, 2847, 10405, 190137, 580816, 770953, 1351769, 2122722, 18333545, 20456267, 38789812, 59246079, 216528049, 3956750961, 12086780932, 16043531893, 28130312825, 44173844718, 381521070569, 425694915287, 807215985856

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 (0,0,0,0,0,0,0,0,0,20810,0,0,0,0,0,0,0,0,0,-1).

Cf. A010537, A041153.

Numerator[Convergents[Sqrt[86], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^19 -9*x^18 +28*x^17 -37*x^16 +65*x^15 -102*x^14 +881*x^13 -983*x^12 +1864*x^11 -2847*x^10 -10405*x^9 -2847*x^8 -1864*x^7 -983*x^6 -881*x^5 -102*x^4 -65*x^3 -37*x^2 -28*x -9) / (x^20 -20810*x^10 +1). - Colin Barker, Nov 10 2013

More terms from Colin Barker, Nov 10 2013

A041153 Denominators of continued fraction convergents to sqrt(86).

Original entry on oeis.org

1, 3, 4, 7, 11, 95, 106, 201, 307, 1122, 20503, 62631, 83134, 145765, 228899, 1976957, 2205856, 4182813, 6388669, 23348820, 426667429, 1303351107, 1730018536, 3033369643, 4763388179, 41140475075, 45903863254, 87044338329, 132948201583, 485888943078

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 (0,0,0,0,0,0,0,0,0,20810,0,0,0,0,0,0,0,0,0,-1).

Cf. A041152, A010159, A020843, A010537.

I:=[1, 3, 4, 7, 11, 95, 106, 201, 307, 1122, 20503, 62631, 83134, 145765, 228899, 1976957, 2205856, 4182813, 6388669, 23348820]; [n le 20 select I[n] else 20810*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 12 2013

Denominator/@Convergents[Sqrt[86], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^18 - 3 x^17 + 4 x^16 - 7 x^15 + 11 x^14 - 95 x^13 + 106 x^12 - 201 x^11 + 307 x^10 - 1122 x^9 - 307 x^8 - 201 x^7 - 106 x^6 - 95 x^5 - 11 x^4 - 7 x^3 - 4 x^2 - 3 x - 1)/(x^20 - 20810 x^10 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 12 2013 *)

G.f.: -(x^18 -3*x^17 +4*x^16 -7*x^15 +11*x^14 -95*x^13 +106*x^12 -201*x^11 +307*x^10 -1122*x^9 -307*x^8 -201*x^7 -106*x^6 -95*x^5 -11*x^4 -7*x^3 -4*x^2 -3*x-1) / (x^20-20810*x^10+1). - Colin Barker, Nov 13 2013

a(n) = 20810*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 12 2013

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A248309 Egyptian fraction representation of sqrt(86) (A010537) using a greedy function.

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A010159 Continued fraction for sqrt(86).

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A041152 Numerators of continued fraction convergents to sqrt(86).

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A041153 Denominators of continued fraction convergents to sqrt(86).

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