A010519 -id:A010519 - cp's OEIS frontend

A248291 Egyptian fraction representation of sqrt(67) (A010519) using a greedy function.

8, 6, 54, 5968, 37928283, 14186508539132240, 215574431124169048574472920051105, 619113864242566215185357331731644567622871533734575552668037746157, 1026704635586993757466869990798845550899476775104786232072062922961543188349830119350020024351935013557637621452049626418228548257237

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[ 67]]

A010147 Continued fraction for sqrt(67).

Original entry on oeis.org

8, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1

Offset: 0

N. J. A. Sloane

			8.185352771872449969953703724... = 8 + 1/(5 + 1/(2 + 1/(1 + 1/(1 + ...)))). - _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, 0, 0, 1).

Cf. A010519 Decimal expansion. - Harry J. Smith, Jun 08 2009

ContinuedFraction[Sqrt[67],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,5,2,1,1,7,1,1,2,5}] (* Harvey P. Dale, Jun 02 2025 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(67)); for (n=0, 20000, write("b010147.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

A176443 Decimal expansion of sqrt(469).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of sqrt(469) is A040447.

			sqrt(469) = 21.65640782770771520178...

Cf. A010465 (decimal expansion of sqrt(7)), A010519 (decimal expansion of sqrt(67)), A176442 (decimal expansion of (21+sqrt(469))/6), A040447.

RealDigits[Sqrt[469],10,120][[1]] (* Harvey P. Dale, May 28 2025 *)

A041116 Numerators of continued fraction convergents to sqrt(67).

Original entry on oeis.org

8, 41, 90, 131, 221, 1678, 1899, 3577, 9053, 48842, 790525, 4001467, 8793459, 12794926, 21588385, 163913621, 185502006, 349415627, 884333260, 4771081927, 77221644092, 390879302387, 858980248866, 1249859551253, 2108839800119, 16011738152086, 18120577952205

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 (0,0,0,0,0,0,0,0,0,97684,0,0,0,0,0,0,0,0,0,-1).

Cf. A010519, A041117.

Numerator[Convergents[Sqrt[67], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^19 -8*x^18 +41*x^17 -90*x^16 +131*x^15 -221*x^14 +1678*x^13 -1899*x^12 +3577*x^11 -9053*x^10 -48842*x^9 -9053*x^8 -3577*x^7 -1899*x^6 -1678*x^5 -221*x^4 -131*x^3 -90*x^2 -41*x -8) / (x^20 -97684*x^10 +1). - Colin Barker, Nov 10 2013

More terms from Colin Barker, Nov 10 2013

A041117 Denominators of continued fraction convergents to sqrt(67).

Original entry on oeis.org

1, 5, 11, 16, 27, 205, 232, 437, 1106, 5967, 96578, 488857, 1074292, 1563149, 2637441, 20025236, 22662677, 42687913, 108038503, 582880428, 9434125351, 47753507183, 104941139717, 152694646900, 257635786617, 1956145153219, 2213780939836, 4169926093055

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 (0,0,0,0,0,0,0,0,0,97684,0,0,0,0,0,0,0,0,0,-1).

Cf. A041116, A010147, A020824, A010519.

I:=[1, 5, 11, 16, 27, 205, 232, 437, 1106, 5967, 96578, 488857, 1074292, 1563149, 2637441, 20025236, 22662677, 42687913, 108038503, 582880428]; [n le 20 select I[n] else 97684*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[67],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Denominator[Convergents[Sqrt[67], 30]] (* Harvey P. Dale, Oct 03 2012 *)
CoefficientList[Series[-(x^18 - 5 x^17 + 11 x^16 - 16 x^15 + 27 x^14 - 205 x^13 + 232 x^12 - 437 x^11 + 1106 x^10 - 5967 x^9 - 1106 x^8 - 437 x^7 - 232 x^6 - 205 x^5 - 27 x^4 - 16 x^3 - 11 x^2 - 5 x - 1)/(x^20 - 97684 x^10 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

G.f.: -(x^18 -5*x^17 +11*x^16 -16*x^15 +27*x^14 -205*x^13 +232*x^12 -437*x^11 +1106*x^10 -5967*x^9 -1106*x^8 -437*x^7 -232*x^6 -205*x^5 -27*x^4 -16*x^3 -11*x^2 -5*x -1) / (x^20 -97684*x^10 +1). - Colin Barker, Nov 13 2013

a(n) = 97684*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 13 2013

cp's OEIS Frontend

A248291 Egyptian fraction representation of sqrt(67) (A010519) using a greedy function.

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A010147 Continued fraction for sqrt(67).

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A176443 Decimal expansion of sqrt(469).

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A041116 Numerators of continued fraction convergents to sqrt(67).

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A041117 Denominators of continued fraction convergents to sqrt(67).

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Magma

Mathematica

Formula

Extensions