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A108716
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a(n) = tan(Pi/14)^(-2n) + tan(3*Pi/14)^(-2n) + tan(5*Pi/14)^(-2n).
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9
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3, 21, 371, 7077, 135779, 2606261, 50028755, 960335173, 18434276035, 353858266965, 6792546291251, 130387472704741, 2502874814474531, 48044357383337973, 922243598852422035, 17703083191185355397
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OFFSET
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0,1
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COMMENTS
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The Berndt-type sequence number 11 for the argument 2Pi/7 defined by the relation a(n) = t(1)^(2*n) + t(2)^(2*n) + t(4)^(2*n) = (-sqrt(7) + 4*s(1))^(2*n) + (-sqrt(7) + 4*s(2))^(2*n) + (-sqrt(7) + 4*s(4))^(2*n), where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) (the respective sum with odd powers are discussed in A215794). See also A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215694, A215695, A215828 and especially A215575, where a(n)=B(2n) for the function B(n) defined in the comments. - Roman Witula, Aug 23 2012
The sequence a(n+1)/a(n) is increasing and convergent to (t(2))^2 = 19,195669... (we note that the sequence A215794(n+1)/A215794(n) is decreasing and convergent to the same limit). - Roman Witula, Aug 24 2012
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REFERENCES
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R. Witula, D. Slota, New Ramanujan-type formulas and quasi-Fibonacci numbers of order 7, J. Integer Seq., 10 (2007), Article 07.5.6.
R. Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, 12 (2009), Article 09.8.5.
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LINKS
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Table of n, a(n) for n=0..15.
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FORMULA
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a(n) = 7^n*A(2n), where A(n) := A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, and A(2)=3. - see Witula-Slota's (Section 6) and Witula's (Remark 11) papers for the proofs and details. In these papers A(n) denotes the value of big omega function with index n for the argument 2i/sqrt(7) (see also A215512). - Roman Witula, Aug 23 2012
Conjecture: a(n) = 21*a(n-1)-35*a(n-2)+7*a(n-3). G.f.: -(35*x^2-42*x+3) / (7*x^3-35*x^2+21*x-1). - Colin Barker, Jun 01 2013
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MATHEMATICA
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Table[ Round[ Cot[Pi/14]^(2n) + Cot[3Pi/14]^(2n) + Cot[5Pi/14]^(2n)], {n, 0, 12}] (* Robert G. Wilson v, Jun 21 2005 *)
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CROSSREFS
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Sequence in context: A083228 A186271 A101389 * A084620 A120603 A139224
Adjacent sequences: A108713 A108714 A108715 * A108717 A108718 A108719
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KEYWORD
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nonn
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AUTHOR
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Philippe DELEHAM, Jun 20 2005
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EXTENSIONS
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More terms from Robert G. Wilson v, Jun 21 2005
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STATUS
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approved
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