Search: a019426 -id:a019426
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A019425
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Continued fraction for tan(1/2).
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+10
11
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0, 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140, 1, 144, 1, 148, 1, 152, 1, 156, 1
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OFFSET
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0,4
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COMMENTS
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The simple continued fraction expansion for tan(1/2) may be derived by setting z = 1/2 in Lambert's continued fraction tan(z) = z/(1 - z^2/(3 - z^2/(5 - ... ))) and, after using an equivalence transformation, making repeated use of the identity 1/(n - 1/m) = 1/((n - 1) + 1/(1 + 1/(m - 1))).
The same approach produces the simple continued fraction expansions for the numbers tan(1/n), n*tan(1/n) and 1/n*tan(1/n) for n = 1,2,3,.... [added Oct 03 2023: and, even more generally, the simple continued fraction expansions for the numbers d*tan(1/n) and 1/d*tan(1/n), where d divides n. See A019429 for an example]. (End)
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LINKS
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FORMULA
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a(n) = n - 1/2 - (n-3/2)*(-1)^n + binomial(1,n) - 2*binomial(0,n). - Paul Barry, Oct 25 2007
a(n) = 2*a(n-2) - a(n-4), n>=6.
G.f.: (x + x^2 + 2*x^3 - x^4 + x^5)/(1-x^2)^2. (End)
Related simple continued fraction expansions:
2*tan(1/2) = [1, 10, 1, 3, 1, 26, 1, 7, 1, 42, 1, 11, 1, 58, 1, 15, 1, 74, 1, 19, 1, 90, ...]
(1/2)*tan(1/2) = [0; 3, 1, 1, 1, 18, 1, 5, 1, 34, 1, 9, 1, 50, 1, 13, 1, 66, 1, 17, 1, 82, ...]. (End)
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EXAMPLE
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0.546302489843790513255179465... = 0 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + ...)))). - Harry J. Smith, Jun 13 2009
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MAPLE
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a := n -> if n < 2 then n else ifelse(irem(n, 2) = 0, 1, 2*n - 2) fi:
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MATHEMATICA
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Join[{0, 1}, LinearRecurrence[{0, 2, 0, -1}, {1, 4, 1, 8}, 100]] (* Vincenzo Librandi, Jan 03 2016 *)
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PROG
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(PARI) { allocatemem(932245000); default(realprecision, 85000); x=contfrac(tan(1/2)); for (n=0, 20000, write("b019425.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009
(Magma) [0, 1] cat [n-1/2-(n-3/2)*(-1)^n+Binomial(1, n)- 2*Binomial(0, n): n in [2..80]]; // Vincenzo Librandi, Jan 03 2016
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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A019429
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Continued fraction for tan(1/6).
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+10
4
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0, 5, 1, 16, 1, 28, 1, 40, 1, 52, 1, 64, 1, 76, 1, 88, 1, 100, 1, 112, 1, 124, 1, 136, 1, 148, 1, 160, 1, 172, 1, 184, 1, 196, 1, 208, 1, 220, 1, 232, 1, 244, 1, 256, 1, 268, 1, 280, 1, 292, 1, 304, 1, 316, 1, 328, 1, 340, 1, 352, 1, 364, 1, 376, 1, 388, 1, 400, 1, 412, 1, 424, 1, 436, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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Conjecture: a(n) = (-1+3*(-1)^n-6*(-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+6*x^2+x+5) / ((x-1)^2*(x+1)^2). - Colin Barker, May 28 2013
a(2*n) = 1 and a(2*n+1) = 12*n + 4, both for n >= 1.
The above conjectures are correct. The simple continued fraction expansion for tan(1/6) may be derived by setting z = 1/6 in Lambert's continued fraction tan(z) = z/(1 - z^2/(3 - z^2/(5 - ... ))) and then, after using an equivalence transformation, making repeated use of the identity 1/(n - 1/m) = 1/((n - 1) + 1/(1 + 1/(m - 1))).
A similar approach produces the related simple continued fraction expansions
2*tan(1/6) = [0, 2, 1, 34, 1, 13, 1, 82, 1, 25, 1, 130, 1, 37, 1, 178, 1, 49, ...], with denominators c(2*n) = 1, c(4*n+1) = 12*n + 1, both for n >= 1, and c(4*n+3) = 48*n + 34 for n >= 0;
3*tan(1/6) = [0; 1, 1, 52, 1, 8, 1, 124, 1, 16, 1, 196, 1, 24, 1, 268, 1, 32, ...];
6*tan(1/6) = [1; 106, 1, 3, 1, 250, 1, 7, 1, 394, 1, 11, 1, 538, 1, 15, 1, 682,..];
(1/2)*tan(1/6) = [0, 11, 1, 7, 1, 58, 1, 19, 1, 106, 1, 31, 1, 154, 1, 43, 1, ...];
(1/3)*tan(1/6) = [0, 17, 1, 4, 1, 88, 1, 12, 1, 160, 1, 20, 1, 232, 1, 28, 1, ...];
(1/6)*tan(1/6) = [0, 35, 1, 1, 1, 178, 1, 5, 1, 322, 1, 9, 1, 466, 1, 13, 1, ...];
(End)
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EXAMPLE
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0.16822721830224246125721608... = 0 + 1/(5 + 1/(1 + 1/(16 + 1/(1 + ...)))). - Harry J. Smith, Jun 13 2009
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MATHEMATICA
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Block[{$MaxExtraPrecision=1000}, ContinuedFraction[Tan[1/6], 100]] (* Harvey P. Dale, May 14 2014 *)
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PROG
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(PARI) { allocatemem(932245000); default(realprecision, 95000); x=contfrac(tan(1/6)); for (n=0, 20000, write("b019429.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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A161012
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Decimal expansion of tan(1/3).
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+10
2
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3, 4, 6, 2, 5, 3, 5, 4, 9, 5, 1, 0, 5, 7, 5, 4, 9, 1, 0, 3, 8, 5, 4, 3, 5, 6, 5, 6, 0, 9, 7, 4, 0, 7, 7, 4, 5, 9, 5, 7, 0, 3, 9, 1, 6, 1, 8, 9, 8, 0, 0, 2, 1, 7, 9, 7, 6, 4, 4, 4, 0, 6, 4, 8, 9, 8, 5, 9, 7, 6, 5, 7, 4, 9, 1, 5, 4, 7, 5, 5, 2, 8, 1, 5, 9, 6, 5, 7, 8, 7, 0, 0, 9, 3, 9, 7, 2, 6, 6, 9, 0, 5, 0, 4, 9
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OFFSET
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0,1
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COMMENTS
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LINKS
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EXAMPLE
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0.346253549510575491038543565609740774595703916189800217976444064898597...
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MATHEMATICA
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RealDigits[Tan[1/3], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)
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PROG
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(PARI) default(realprecision, 20080); x=10*tan(1/3); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b161012.txt", n, " ", d));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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