A010514 -id:A010514 - cp's OEIS frontend

A248286 Egyptian fraction representation of sqrt(61) (A010514) using a greedy function.

7, 2, 4, 17, 702, 607877, 1343651924022, 4320622614714270261311118, 32109651275722538015654226404724112550695835225776, 2887634404082286927710711082091702089862802645035135042777568254515668100623050781361931122852713355

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[ 61]]

A010145 Continued fraction for sqrt(61).

Original entry on oeis.org

7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14

Offset: 0

N. J. A. Sloane

			7.810249675906654394129722735... = 7 + 1/(1 + 1/(4 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 07 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
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. A010514 Decimal expansion. - Harry J. Smith, Jun 07 2009

ContinuedFraction[Sqrt[61],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{7},120,{14,1,4,3,1,2,2,1,3,4,1}] (* Harvey P. Dale, Mar 27 2013 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(61)); for (n=0, 20000, write("b010145.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

A041106 Numerators of continued fraction convergents to sqrt(61).

Original entry on oeis.org

7, 8, 39, 125, 164, 453, 1070, 1523, 5639, 24079, 29718, 440131, 469849, 2319527, 7428430, 9747957, 26924344, 63596645, 90520989, 335159612, 1431159437, 1766319049, 26159626123, 27925945172, 137863406811, 441516165605, 579379572416, 1600275310437

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 (0,0,0,0,0,0,0,0,0,0,59436,0,0,0,0,0,0,0,0,0,0,1).

Cf. A010514, A041107, A010145, A020818.

Numerator[Convergents[Sqrt[61], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^21 -7*x^20 +8*x^19 -39*x^18 +125*x^17 -164*x^16 +453*x^15 -1070*x^14 +1523*x^13 -5639*x^12 +24079*x^11 +29718*x^10 +24079*x^9 +5639*x^8 +1523*x^7 +1070*x^6 +453*x^5 +164*x^4 +125*x^3 +39*x^2 +8*x +7) / (x^22 +59436*x^11 -1). - Colin Barker, Nov 12 2013

More terms from Colin Barker, Nov 12 2013

A041107 Denominators of continued fraction convergents to sqrt(61).

Original entry on oeis.org

1, 1, 5, 16, 21, 58, 137, 195, 722, 3083, 3805, 56353, 60158, 296985, 951113, 1248098, 3447309, 8142716, 11590025, 42912791, 183241189, 226153980, 3349396909, 3575550889, 17651600465, 56530352284, 74181952749, 204894257782, 483970468313, 688864726095

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 (0,0,0,0,0,0,0,0,0,0,59436,0,0,0,0,0,0,0,0,0,0,1).

Cf. A041106, A010145, A020818, A010514.

I:=[1, 1, 5, 16, 21, 58, 137, 195, 722, 3083, 3805, 56353, 60158, 296985, 951113, 1248098, 3447309, 8142716, 11590025, 42912791, 183241189, 226153980]; [n le 22 select I[n] else 59436*Self(n-11)+Self(n-22): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Denominator[Convergents[Sqrt[61], 30]] (* Vincenzo Librandi, Dec 11 2013 *)

G.f.: -(x^20 -x^19 +5*x^18 -16*x^17 +21*x^16 -58*x^15 +137*x^14 -195*x^13 +722*x^12 -3083*x^11 +3805*x^10 +3083*x^9 +722*x^8 +195*x^7 +137*x^6 +58*x^5 +21*x^4 +16*x^3 +5*x^2 +x +1) / (x^22 +59436*x^11 -1). - Colin Barker, Nov 12 2013

a(n) = 59436*a(n-11) + a(n-22). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

cp's OEIS Frontend

A248286 Egyptian fraction representation of sqrt(61) (A010514) using a greedy function.

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A010145 Continued fraction for sqrt(61).

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A041106 Numerators of continued fraction convergents to sqrt(61).

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A041107 Denominators of continued fraction convergents to sqrt(61).

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Magma

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