A019430 -id:A019430 - cp's OEIS frontend

A019426 Continued fraction for tan(1/3).

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

Offset: 0

David W. Wilson

The simple continued fraction expansion of 3*tan(1/3) is [1; 25, 1, 3, 1, 61, 1, 7, 1, 97, 1, 11, 1, ..., 36*n + 25, 1, 4*n + 3, 1, ...], while the simple continued fraction expansion of (1/3)*tan(1/3) is [0; 8, 1, 1, 1, 43, 1, 5, 1, 79, 1, 9, 1, 115, 1, 13, 1, ..., 36*n + 7, 1, 4*n + 1, 1, ...]. See my comment in A019425. - Peter Bala, Sep 30 2023

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

Harry J. Smith, Table of n, a(n) for n = 0..20000
Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, Continued Fractions with Predictable Patterns and Transcendental Numbers, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.4.
G. Xiao, Contfrac
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

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).

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

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

{ 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

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)

A126980 a(n) = 14*n + 47.

Original entry on oeis.org

47, 61, 75, 89, 103, 117, 131, 145, 159, 173, 187, 201, 215, 229, 243, 257, 271, 285, 299, 313, 327, 341, 355, 369, 383, 397, 411, 425, 439, 453, 467, 481, 495, 509, 523, 537, 551, 565, 579, 593, 607, 621, 635, 649, 663, 677, 691, 705, 719, 733, 747

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Superhighway created by 'LQTL Ant' L90R135L90R135 from iteration 47 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the turn angle (in degrees) at each iteration.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, 2000.
S. Wolfram, A New Kind of Science, 1st ed. Il.: Wolfram Media Inc., 2002.

Diana Mecum, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017041, A019430, A094867, A126978, A126979.

[14*n + 47: n in [0..60] ]; // Vincenzo Librandi, Jul 18 2011

Table[14*n + 47, {n, 0, 60}] (* Stefan Steinerberger, Jun 17 2007 *)
LinearRecurrence[{2,-1}, {47,61}, 50] (* G. C. Greubel, May 30 2016 *)

From Chai Wah Wu, May 30 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (-33*x + 47)/(x - 1)^2. (End)
E.g.f.: (47 + 14*x)*exp(x). - G. C. Greubel, May 30 2016

More terms from Stefan Steinerberger and Diana L. Mecum, Jun 17 2007

A161016 Decimal expansion of tan(1/7).

Original entry on oeis.org

1, 4, 3, 8, 3, 6, 9, 5, 9, 4, 3, 6, 1, 9, 0, 9, 3, 5, 2, 8, 0, 0, 3, 0, 5, 9, 9, 1, 3, 5, 6, 2, 3, 3, 4, 4, 9, 8, 7, 5, 6, 8, 1, 6, 9, 6, 7, 3, 8, 8, 8, 5, 3, 9, 3, 3, 0, 8, 7, 1, 0, 0, 1, 6, 5, 3, 1, 8, 6, 4, 4, 4, 9, 9, 4, 6, 1, 4, 5, 3, 6, 3, 4, 8, 4, 9, 8, 7, 3, 6, 8, 3, 9, 2, 6, 8, 2, 6, 8, 4, 3, 9, 5, 4, 5

Offset: 0

Harry J. Smith, Jun 14 2009

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.143836959436190935280030599135623344987568169673888539330871001653186...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A019430 Continued fraction.

RealDigits[Tan[1/7],10,120][[1]] (* Harvey P. Dale, Dec 11 2016 *)

default(realprecision, 20080); x=10*tan(1/7); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b161016.txt", n, " ", d));

cp's OEIS Frontend

A019426 Continued fraction for tan(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A126980 a(n) = 14*n + 47.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A161016 Decimal expansion of tan(1/7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI