A019432 Continued fraction for tan(1/9).

0, 8, 1, 25, 1, 43, 1, 61, 1, 79, 1, 97, 1, 115, 1, 133, 1, 151, 1, 169, 1, 187, 1, 205, 1, 223, 1, 241, 1, 259, 1, 277, 1, 295, 1, 313, 1, 331, 1, 349, 1, 367, 1, 385, 1, 403, 1, 421, 1, 439, 1, 457, 1, 475, 1, 493, 1, 511, 1, 529, 1, 547, 1, 565, 1, 583, 1, 601, 1, 619, 1, 637, 1, 655

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• cofr
• easy

Comments

The odd-indexed terms from and after a(3) are equal to 18n+7. - Harvey P. Dale, Sep 26 2021

Examples

			0.11157062783380058372650480... = 0 + 1/(8 + 1/(1 + 1/(25 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 14 2009
		

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).

Crossrefs

Cf. A161018 (decimal expansion), A019425 through A019433.

Programs

Mathematica

Block[{$MaxExtraPrecision=1000}, ContinuedFraction[Tan[1/9],100]] (* or *) LinearRecurrence[{0,2,0,-1},{0,8,1,25,1,43},80] (* Harvey P. Dale, Sep 26 2021 *)

PARI

{ allocatemem(932245000); default(realprecision, 98000); x=contfrac(tan(1/9)); for (n=0, 20000, write("b019432.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 14 2009

PARI

Vec(x*(x^4-x^3+9*x^2+x+8)/((x-1)^2*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 08 2013

Formula

From Colin Barker, Sep 08 2013: (Start)

a(n) = (-1+3*(-1)^n-9*(-1+(-1)^n)*n)/2 for n>1.

a(n) = 2*a(n-2)-a(n-4) for n>5.

G.f.: x*(x^4-x^3+9*x^2+x+8) / ((x-1)^2*(x+1)^2). (End)