OFFSET
0,1
COMMENTS
The Berndt-type sequence number 11 for the argument 2*Pi/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 converges to the same limit). - Roman Witula, Aug 24 2012
Let L(p) be the total length of all sides and diagonals of a regular p-sided polygon inscribed in a unit circle. Then (L(p)/p)^2 = cot(Pi/(2p))^2 is the largest root of the equation: C(p,k)-C(p,2+k)*x+C(p,4+k)*x^2-C(p,6+k)*x^3+ ... +(-1)^q*x^q = 0, where k=1 if p is odd, k=0 if p is even, q = floor(p/2), and where C denotes the binomial coefficient. The complete set of roots is: x(i) = cot((2*i-1)*Pi/(2p))^2, i=1,2,...,q. Then a(n) = x(1)^n+x(2)^n+...x(q)^n for p=7. - Seppo Mustonen, Mar 25 2014
Sum_{k=1..(m-1)/2} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM link and formula); this sequence is the particular case m = 7. All terms are odd. - Bernard Schott, Apr 22 2022
LINKS
Robert Israel, Table of n, a(n) for n = 0..700
Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
Seppo Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations (2013).
Seppo Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations [Local copy]
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Index entries for linear recurrences with constant coefficients, signature (21,-35,7).
FORMULA
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 the big omega function with index n for the argument 2*i/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
To verify conjecture, note that the roots of 7*x^3-35*x^2+21*x-1 are tan(Pi/14)^2, tan(3*Pi/14)^2 and tan(5*Pi/14)^2. - Robert Israel, Aug 23 2015
E.g.f.: exp((tan(Pi/7))^2*x) + exp((cot(Pi/14))^2*x) + exp((cot(3*Pi/14))^2*x). - G. C. Greubel, Aug 22 2015
a(n) = (tan(1*Pi/7))^(2*n) + (tan(2*Pi/7))^(2*n) + (tan(3*Pi/7))^(2*n). - Bernard Schott, Apr 22 2022
MAPLE
A:= gfun:-rectoproc({-a(n+3)+21*a(n+2)-35*a(n+1)+7*a(n), a(0) = 3, a(1) = 21, a(2) = 371}, a(n), remember):
seq(A(n), n=0..20); # Robert Israel, Aug 23 2015
MATHEMATICA
Table[ Round[ Cot[Pi/14]^(2n) + Cot[3Pi/14]^(2n) + Cot[5Pi/14]^(2n)], {n, 0, 12}] (* Robert G. Wilson v, Jun 21 2005 *)
RecurrenceTable[{a[0]== 3, a[1]== 21, a[2]==371, a[n]== 21*a[n-1] - 35*a[n-2] + 7*a[n-3]}, a, {n, 30}] (* G. C. Greubel, Aug 22 2015 *)
PROG
(PARI) a(n)=round(tan(Pi/14)^(-2*n) + tan(3*Pi/14)^(-2*n) + tan(5*Pi/14)^(-2*n)); \\ Anders Hellström, Aug 22 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Jun 20 2005
EXTENSIONS
More terms from Robert G. Wilson v, Jun 21 2005
STATUS
approved