A010498 -id:A010498 - cp's OEIS frontend

A248270 Egyptian fraction representation of sqrt(44) (A010498) using a greedy function.

6, 2, 8, 122, 18919, 402739144, 764123173937021975, 2148666191962903360885805290461855276, 8622580654686644746427953833014483269744901669599325824509666827330296874

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Cf. A010498.
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[ 44]]

A040037 Continued fraction for sqrt(44).

Original entry on oeis.org

6, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane

			6.633249580710799698229865473... = 6 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, Jun 05 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

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

Cf. A010498 (decimal expansion).

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[44],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},80,{12,1,1,1,2,1,1,1}] (* Harvey P. Dale, Apr 02 2013 *)

{ allocatemem(932245000); default(realprecision, 14000); x=contfrac(sqrt(44)); for (n=0, 20000, write("b040037.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 1, a(4) = 2, a(2^e) = 12 for e >= 3, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 5/2^(3*s-1) + 1/4^s). (End)
G.f.: (6 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 6*x^8)/(1 - x^8). - Stefano Spezia, Jul 27 2025

A041074 Numerators of continued fraction convergents to sqrt(44).

Original entry on oeis.org

6, 7, 13, 20, 53, 73, 126, 199, 2514, 2713, 5227, 7940, 21107, 29047, 50154, 79201, 1000566, 1079767, 2080333, 3160100, 8400533, 11560633, 19961166, 31521799, 398222754, 429744553, 827967307, 1257711860

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 (0,0,0,0,0,0,0,398,0,0,0,0,0,0,0,-1).

Cf. A041075, A010498.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[44],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Numerator[Convergents[Sqrt[44], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^15 -6*x^14 +7*x^13 -13*x^12 +20*x^11 -53*x^10 +73*x^9 -126*x^8 -199*x^7 -126*x^6 -73*x^5 -53*x^4 -20*x^3 -13*x^2 -7*x -6) / ((x^8 -20*x^4 +1)*(x^8 +20*x^4 +1)). - Colin Barker, Nov 04 2013

A041075 Denominators of continued fraction convergents to sqrt(44).

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 19, 30, 379, 409, 788, 1197, 3182, 4379, 7561, 11940, 150841, 162781, 313622, 476403, 1266428, 1742831, 3009259, 4752090, 60034339, 64786429, 124820768, 189607197, 504035162, 693642359, 1197677521, 1891319880, 23893516081, 25784835961

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 (0,0,0,0,0,0,0,398,0,0,0,0,0,0,0,-1).

Cf. A010498, A041074.

I:=[1, 1, 2, 3, 8, 11, 19, 30, 379, 409, 788, 1197, 3182, 4379, 7561, 11940]; [n le 16 select I[n] else 398*Self(n-8)-Self(n-16): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[44],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011 *)
Denominator[Convergents[Sqrt[44], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,398,0,0,0,0,0,0,0,-1},{1,1,2,3,8,11,19,30,379,409,788,1197,3182,4379,7561,11940},40] (* Harvey P. Dale, Feb 12 2025 *)

G.f.: -(x^2-x-1)*(x^4+3*x^2+1)*(x^8+10*x^4+1) / ((x^8-20*x^4+1)*(x^8+20*x^4+1)). - Colin Barker, Nov 12 2013

a(n) = 398*a(n-8) - a(n-16). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

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A248270 Egyptian fraction representation of sqrt(44) (A010498) using a greedy function.

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A040037 Continued fraction for sqrt(44).

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A041074 Numerators of continued fraction convergents to sqrt(44).

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A041075 Denominators of continued fraction convergents to sqrt(44).

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