A041052 -id:A041052 - cp's OEIS frontend

A041053 Denominators of continued fraction convergents to sqrt(32).

1, 1, 2, 3, 32, 35, 67, 102, 1087, 1189, 2276, 3465, 36926, 40391, 77317, 117708, 1254397, 1372105, 2626502, 3998607, 42612572, 46611179, 89223751, 135834930, 1447573051, 1583407981, 3030981032, 4614389013

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,34,0,0,0,-1).

Cf. A010487, A041052.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[32],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Denominator[Convergents[Sqrt[32],30]] (* or *) LinearRecurrence[{0,0,0,34,0,0,0,-1},{1,1,2,3,32,35,67,102},30] (* Harvey P. Dale, Jun 15 2015 *)

G.f.: (1+x+2*x^2+3*x^3-2*x^4+x^5-x^6)/(1-34*x^4+x^8). [Colin Barker, Mar 13 2012]

a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=32, a(5)=35, a(6)=67, a(7)=102, a(n)=34*a(n-4)-a(n-8). - Harvey P. Dale, Jun 15 2015

A010130 Continued fraction for sqrt(32).

Original entry on oeis.org

5, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10

Offset: 0

N. J. A. Sloane

			5.65685424949238019520675489... = 5 + 1/(1 + 1/(1 + 1/(1 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 04 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 (0,0,0,1).

Cf. A010487 (decimal expansion).

Cf. A041052/A041053 (convergents), A248259 (Egyptian fraction).

ContinuedFraction[Sqrt[32],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{5},100,{10,1,1,1}] (* Harvey P. Dale, Aug 20 2014 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(32)); for (n=0, 20000, write("b010130.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 1, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 9/4^s). (End)
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (5 + x + x^2 + x^3 + 5*x^4)/((1 - x)*(1 + x + x^2 + x^3)).
a(n) = a(n-4), n > 4. (End)

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A041053 Denominators of continued fraction convergents to sqrt(32).

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