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A019433 Continued fraction for tan(1/10). 9
0, 9, 1, 28, 1, 48, 1, 68, 1, 88, 1, 108, 1, 128, 1, 148, 1, 168, 1, 188, 1, 208, 1, 228, 1, 248, 1, 268, 1, 288, 1, 308, 1, 328, 1, 348, 1, 368, 1, 388, 1, 408, 1, 428, 1, 448, 1, 468, 1, 488, 1, 508, 1, 528, 1, 548, 1, 568, 1, 588, 1, 608, 1, 628, 1, 648, 1, 668, 1, 688, 1, 708, 1, 728 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
From _Peter Bala_, Oct 04 2023: (Start)
Related simple continued fractions expansions (see my comments in A019425):
tan(1/(10*k)) = [0; 10*k - 1, 1, 30*k - 2, 1, 50*k - 2, 1, 70*k - 2, 1, 90*k - 2, 1, ...] for k >= 1.
If d is a divisor of 10 with d*d' = 10 then the simple continued fraction expansion of d*tan(1/10) begins [0; d' - 1, 1, 30*d - 2, 1, 5*d' - 2, 1, 70*d - 2, 1, 9*d' - 2, 1, 110*d - 2, 1, 13*d' - 2, ...], while the simple continued fraction expansion of (1/d)*tan(1/10) begins [ 0; 10*d - 1, 1, 3*d'- 2, 1, 50*d - 2, 1, 7*d' - 2, 1, 90*d - 2, 1, 11*d' - 2, 1, 130*d - 2, ...]. (End)
LINKS
G. Xiao, Contfrac
FORMULA
From _Colin Barker_, Sep 08 2013: (Start)
a(n) = -1/2+(3*(-1)^n)/2+5*n-5*(-1)^n*n for n>1.
a(n) = 2*a(n-2)-a(n-4) for n>5.
G.f.: x*(x^4-x^3+10*x^2+x+9) / ((x-1)^2*(x+1)^2). (End)
EXAMPLE
0.10033467208545054505808004... = 0 + 1/(9 + 1/(1 + 1/(28 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 14 2009
MATHEMATICA
LinearRecurrence[{0, 2, 0, -1}, {0, 9, 1, 28, 1, 48}, 80] (* or *) Join[{0, 9}, Riffle[NestList[20+#&, 28, 40], 1, {1, -1, 2}]] (* _Harvey P. Dale_, Jul 23 2023 *)
PROG
(PARI) { allocatemem(932245000); default(realprecision, 99000); x=contfrac(tan(1/10)); for (n=0, 20000, write("b019433.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 14 2009
(PARI) Vec(x*(x^4-x^3+10*x^2+x+9)/((x-1)^2*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Sep 08 2013
CROSSREFS
Cf. A161019 (decimal expansion), A019425 through A019432.
Sequence in context: A202006 A195278 A092477 * A276634 A050312 A295575
KEYWORD
nonn,cofr,easy
AUTHOR
_David W. Wilson_
STATUS
approved

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Last modified May 29 04:26 EDT 2024. Contains 372921 sequences. (Running on oeis4.)