OFFSET
0,3
COMMENTS
The Berndt-type sequence number 10 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For additional informations and particularly connections with another sequences of trigonometric nature - see comments to A215512 (a(n) is equal to the sequence c(n) in these comments) and Witula-Slota's reference (Section 3).
The following summation formula hold true (see comments to A215512): Sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) + 1, n=3,4,...
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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 (5,-6,1).
FORMULA
sqrt(7)*a(n) = s(1)*c(1)^(2*n) + s(2)*c(2)^(2*n) + s(4)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
G.f.: (1-5*x+4*x^2)/(1-5*x+6*x^2-x^3).
EXAMPLE
We have a(8)=3*a(7)+3*a(5)-6*a(2) and a(9)=3*a(8)+3*a(6)-3*a(4)-a(1).
MATHEMATICA
LinearRecurrence[{5, -6, 1}, {1, 0, -2}, 50]
PROG
(PARI) x='x+O('x^30); Vec((1-5*x+4*x^2)/(1-5*x+6*x^2-x^3)) \\ G. C. Greubel, Apr 25 2018
(Magma) I:=[1, 0, -2]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 25 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Aug 21 2012
STATUS
approved