OFFSET
0,2
COMMENTS
The Berndt-type sequence number 3 for the argument 2Pi/7 (see A215007 and A215008 for the respective sequences numbers 1 and 2) is defined by the following relations: sqrt(7) *a(n) = s(1)*s(2)^(2n) + s(2)*s(4)^(2n) + s(4)*s(1)^(2n) = s(4)*s(1)^(2n) + s(1)*s(2)^(2n) + s(2)*s(4)^(2n), where s(j) := 2*sin(2*Pi*j/7).
REFERENCES
R. Witula, Complex numbers, Polynomials and Fractial Partial Decompositions, T.3, Silesian Technical University Press, Gliwice 2010 (in Polish).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
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 (7,-14,7).
FORMULA
G.f.: (1-5*x+7*x^2)/(1-7*x+14*x^2-7*x^3).
MATHEMATICA
LinearRecurrence[{7, -14, 7}, {1, 2, 7}, 40]
PROG
(PARI) Vec((1-5*x+7*x^2)/(1-7*x+14*x^2-7*x^3) + O(x^30)) \\ Michel Marcus, Apr 19 2016
(Magma) I:=[1, 2, 7]; [n le 3 select I[n] else 7*Self(n-1) - 14*Self(n-2) + 7*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Aug 04 2012
STATUS
approved