A010498 -id:A010498

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A248270

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..8.

MATHEMATICA

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[ Sq

CROSSREFS

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Oct 04 2014

STATUS

approved

A040037

Continued fraction for sqrt(44).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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).

FORMULA

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)

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) { 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

CROSSREFS

Cf. A010498 (decimal expansion).

KEYWORD

nonn,cofr,easy,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

A041074

Numerators of continued fraction convergents to sqrt(44).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A041075, A010498.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

A041075

Denominators of continued fraction convergents to sqrt(44).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

PROG

(Magma) 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

CROSSREFS

Cf. A010498, A041074.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 12 2013

STATUS

approved

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