OFFSET
0,2
COMMENTS
The Berndt-type sequence number 12 for the argument 2Pi/7
defined by the relation sqrt(7)*a(n) = t(1)^(2*n+1) + t(2)^(2*n+1) + t(4)^(2*n+1) = (-sqrt(7) + 4*s(1))^(2*n+1) + (-sqrt(7) + 4*s(2))^(2*n+1) + (-sqrt(7) + 4*s(4))^(2*n+1), where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) (the respective sum with even powers in A108716 are given, see also A215828). We note that sqrt(7)*a(n) = B(2*n+1), where B(n) is defined in the comments to A215575. From Witula-Slota's (Section 6) and Witula's (Remark 11) papers it follows that B(n) is equal to the product (-sqrt(7))^n by the value of big omega function with index n for the argument 2*i/sqrt(7). The last value is equal to A(n). The respective recurrence relation for A(n) from the following decomposition follow (see Witula-Slota's paper for details): (X-1-2*i*d*s(1))*(X-1-2*i*d*s(2))*(X-1- 2*i*d*s(4)) = X^3 - (3+i*sqrt(7))*X^2 + (3+i*2*sqrt(7)*d)*X - (1+i*sqrt(7)*d + i*sqrt(7)*d^3), since the big omega function with index n for the argument d is equal to the sum: (1 + 2*i*d*s(1))^n + (1 + 2*i*d*s(2))^n + (1 + 2*i*d*s(4))^n and it is equal to 3 for n=0, 3 + i*sqrt(7)*d for n=1, and at last 3 + 2*i*sqrt(7)*d - 7*d^2 for n=2.
The sequence a(n+1)/a(n) is decreasing and convergent to (t(2))^2 = 19,195669... Moreover we have floor(a(n+1)/a(n)) = 19 for every n=1,2,...
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5
Index entries for linear recurrences with constant coefficients, signature (21,-35,7).
FORMULA
G.f.: -(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3). [Bruno Berselli, Aug 30 2012]
EXAMPLE
We have -31*sqrt(7) = t(1)^3 + t(2)^3 + t(4)^3.
MATHEMATICA
LinearRecurrence[{21, -35, 7}, {-1, -31, -609}, 17] (* Bruno Berselli, Aug 30 2012 *)
PROG
(Magma) m:=17; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3))); // Bruno Berselli, Aug 30 2012
(Magma) I:=[-1, -31, -609]; [n le 3 select I[n] else 21*Self(n-1)-35*Self(n-2)+7*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Mar 19 2013
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Aug 23 2012
STATUS
approved