A019431 Continued fraction for tan(1/8).

0, 7, 1, 22, 1, 38, 1, 54, 1, 70, 1, 86, 1, 102, 1, 118, 1, 134, 1, 150, 1, 166, 1, 182, 1, 198, 1, 214, 1, 230, 1, 246, 1, 262, 1, 278, 1, 294, 1, 310, 1, 326, 1, 342, 1, 358, 1, 374, 1, 390, 1, 406, 1, 422, 1, 438, 1, 454, 1, 470, 1, 486, 1, 502, 1, 518, 1, 534, 1, 550, 1, 566, 1, 582

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• cofr
• easy

Examples

			0.12565513657513096779267821... = 0 + 1/(7 + 1/(1 + 1/(22 + 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. A161017 (decimal expansion), A019425 through A019433.

Programs

Magma

[0,7] cat [(-1+3*(-1)^n-8*(-1+(-1)^n)*n)/2: n in [2..80]]; // Vincenzo Librandi, Jan 03 2016

Mathematica

Join[{0, 7}, LinearRecurrence[{0, 2, 0, -1}, {1, 22, 1, 38}, 100]] (* Vincenzo Librandi, Jan 03 2016 *)
Block[{$MaxExtraPrecision=1000}, ContinuedFraction[Tan[1/8],100]] (* Harvey P. Dale, Jul 14 2025 *)

PARI

{ allocatemem(932245000); default(realprecision, 97000); x=contfrac(tan(1/8)); for (n=0, 20000, write("b019431.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 14 2009

PARI

Vec(x*(x^4-x^3+8*x^2+x+7)/((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-8*(-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+8*x^2+x+7) / ((x-1)^2*(x+1)^2). (End)