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A019426 Continued fraction for tan(1/3). 2
0, 2, 1, 7, 1, 13, 1, 19, 1, 25, 1, 31, 1, 37, 1, 43, 1, 49, 1, 55, 1, 61, 1, 67, 1, 73, 1, 79, 1, 85, 1, 91, 1, 97, 1, 103, 1, 109, 1, 115, 1, 121, 1, 127, 1, 133, 1, 139, 1, 145, 1, 151, 1, 157, 1, 163, 1, 169, 1, 175, 1, 181, 1, 187, 1, 193, 1, 199, 1, 205, 1, 211, 1, 217, 1, 223, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

G. Xiao, Contfrac

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

FORMULA

From Bruno Berselli, Sep 21 2012: (Start)

G.f.: x*(2+x+3*x^2-x^3+x^4)/(1-x^2)^2.

a(n) = 2*a(n-2)-a(n-4) with n>4, a(0)=0, a(1)=2, a(2)=1, a(3)=7, a(4)=1.

a(n) = 1+3*(1-(-1)^n)*(n-1)/2 with n>1, a(0)=0, a(1)=2.

For k>0: a(2k) = 1, a(4k+1) = 2*a(2k+1)-1 and a(4k+3) = 2*a(2k+1)+5, with a(0)=0, a(1)=2. (End)

EXAMPLE

0.346253549510575491038543565... = 0 + 1/(2 + 1/(1 + 1/(7 + 1/(1 + ...)))). - Harry J. Smith, Jun 13 2009

MATHEMATICA

ContinuedFraction[Tan[1/3], 80] (* Bruno Berselli, Sep 21 2012 *)

PROG

(PARI) { allocatemem(932245000); default(realprecision, 88000); x=contfrac(tan(1/3)); for (n=0, 20000, write("b019426.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009

(MAGMA) [n le 1 select 2*n else 1+3*(1-(-1)^n)*(n-1)/2: n in [0..80]]; // Bruno Berselli, Sep 21 2012

CROSSREFS

Cf. A161012 (decimal expansion of tan(1/3)).

Cf. continued fractions for tan(1/m): A019425 (m=2),  A019427 (m=4), A019428 (m=5), A019429 (m=6), A019430 (m=7), A019431 (m=8), A019432 (m=9), A019433 (m=10), A093178 (m=1).

Sequence in context: A117044 A280691 A092666 * A128747 A205945 A124392

Adjacent sequences:  A019423 A019424 A019425 * A019427 A019428 A019429

KEYWORD

nonn,easy,cofr

AUTHOR

David W. Wilson

STATUS

approved

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Last modified July 15 22:48 EDT 2019. Contains 325061 sequences. (Running on oeis4.)