OFFSET
0,2
COMMENTS
The Berndt-type sequence number 9 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For more connections with another sequences of trigonometric nature see comments to A215512 (a(n) is equal to the sequence b(n) in these comments) and Witula-Slota's reference (Section 3). We note that a(n)=A109682(n) for n=1,2,3,4. Moreover the following summation formula hold true: sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) - 9, for every n=3,4,... - see comments to A215512.
The inverse binomial transform is 1,1, 4, 8, 19, 42, 95,... essentially a shifted, unsigned variant of A215112. - R. J. Mathar, Aug 22 2012
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(4)*c(1)^(2*n) + s(1)*c(2)^(2*n) + s(2)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
G.f.: (1-3*x+3*x^2)/(1-5*x+6*x^2-x^3).
EXAMPLE
We have 10*a(3) = 3*a(4), a(0)+a(1)+3*a(2) = a(3), a(0)+a(2)+3*a(3) = a(4), a(1)+3*a(2)+3*a(4) = a(5), and a(6) = 3*a(5)+3*a(4)-a(1).
MATHEMATICA
LinearRecurrence[{5, -6, 1}, {1, 2, 7}, 50]
PROG
(PARI) Vec((1-3*x+3*x^2)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
(Magma) I:=[1, 2, 7]; [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
nonn,easy
AUTHOR
Roman Witula, Aug 21 2012
STATUS
approved