A010528 -id:A010528 - cp's OEIS frontend

A248300 Egyptian fraction representation of sqrt(76) (A010528) using a greedy function.

8, 2, 5, 57, 3937, 37141276, 4653057274142158, 47471949655200856696698957090199, 11484366883753641302577416484692763851090325557224592536410101596, 1021543423762203659811429437059378653018184069838777743837274072099337791716358457056702326068913163280349514932095292745066307994

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[ 76]]

A010154 Continued fraction for sqrt(76).

Original entry on oeis.org

8, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5

Offset: 0

N. J. A. Sloane

			8.717797887081347104473963967... = 8 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 09 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, 0, 1).

Cf. A010528 Decimal expansion. - Harry J. Smith, Jun 09 2009

ContinuedFraction[Sqrt[76],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},100,{16,1,2,1,1,5,4,5,1,1,2,1}] (* Harvey P. Dale, Jul 06 2021 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(76)); for (n=0, 20000, write("b010154.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 09 2009

A041134 Numerators of continued fraction convergents to sqrt(76).

Original entry on oeis.org

8, 9, 26, 35, 61, 340, 1421, 7445, 8866, 16311, 41488, 57799, 966272, 1024071, 3014414, 4038485, 7052899, 39302980, 164264819, 860627075, 1024891894, 1885518969, 4795929832, 6681448801, 111699110648

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 (0,0,0,0,0,0,0,0,0,0,0,115598,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A010528, A041135.

Numerator[Convergents[Sqrt[76],30]] (* Harvey P. Dale, Aug 24 2011 *)
CoefficientList[Series[- (x^23 - 8 x^22 + 9 x^21 - 26 x^20 + 35 x^19 - 61 x^18 + 340 x^17 - 1421 x^16 + 7445 x^15 - 8866 x^14 + 16311 x^13 - 41488 x^12 - 57799 x^11 - 41488 x^10 - 16311 x^9 - 8866 x^8 - 7445 x^7 - 1421 x^6 - 340 x^5 - 61 x^4 - 35 x^3 - 26 x^2 - 9 x - 8)/((x^12 - 340 x^6 + 1) (x^12 + 340 x^6 + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 26 2013 *)

a(n) = 115598*a(n-12)-a(n-24). G.f.: -(x^23 -8*x^22 +9*x^21 -26*x^20 +35*x^19 -61*x^18 +340*x^17 -1421*x^16 +7445*x^15 -8866*x^14 +16311*x^13 -41488*x^12 -57799*x^11 -41488*x^10 -16311*x^9 -8866*x^8 -7445*x^7 -1421*x^6 -340*x^5 -61*x^4 -35*x^3 -26*x^2 -9*x-8) /( (x^12-340*x^6+1)*(x^12+340*x^6+1) ). [Colin Barker, Jul 19 2012]

Formula corrected by Colin Barker, Jul 24 2012

A041135 Denominators of continued fraction convergents to sqrt(76).

Original entry on oeis.org

1, 1, 3, 4, 7, 39, 163, 854, 1017, 1871, 4759, 6630, 110839, 117469, 345777, 463246, 809023, 4508361, 18842467, 98720696, 117563163, 216283859, 550130881, 766414740, 12812766721, 13579181461, 39971129643, 53550311104, 93521440747, 521157514839, 2178151500103

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 (0,0,0,0,0,0,0,0,0,0,0,115598,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A010528, A041134.

Denominator/@Convergents[Sqrt[76], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)

a(n) = 115598*a(n-12)-a(n-24). G.f.: -(x^22 -x^21 +3*x^20 -4*x^19 +7*x^18 -39*x^17 +163*x^16 -854*x^15 +1017*x^14 -1871*x^13 +4759*x^12 -6630*x^11 -4759*x^10 -1871*x^9 -1017*x^8 -854*x^7 -163*x^6 -39*x^5 -7*x^4 -4*x^3 -3*x^2 -x -1)/((x^12-340*x^6+1)*(x^12+340*x^6+1)). [Colin Barker, Jul 19 2012]

Formula corrected by Colin Barker, Jul 24 2012

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A248300 Egyptian fraction representation of sqrt(76) (A010528) using a greedy function.

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A010154 Continued fraction for sqrt(76).

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A041134 Numerators of continued fraction convergents to sqrt(76).

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A041135 Denominators of continued fraction convergents to sqrt(76).

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Mathematica

Formula

Extensions