A040037 Continued fraction for sqrt(44).

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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

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

References

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

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).

Crossrefs

Cf. A010498 (decimal expansion).

Programs

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 *)

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

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)

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