A215143 - OEIS

%I #39 Sep 08 2022 08:46:03

%S 1,2,7,28,112,441,1715,6615,25382,97069,370440,1411788,5375839,

%T 20458921,77833217,296038498,1125816895,4281011812,16277915640,

%U 61891962377,235320000363,894697938743,3401649302758,12933013979445,49170893188704,186945601728004,710757805310287

%N a(n) = 7*a(n-1) -14*a(n-2) +7*a(n-3), with a(0)=1, a(1)=2, a(2)=7.

%C 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).

%D R. Witula, Complex numbers, Polynomials and Fractial Partial Decompositions, T.3, Silesian Technical University Press, Gliwice 2010 (in Polish).

%H G. C. Greubel, <a href="/A215143/b215143.txt">Table of n, a(n) for n = 0..1000</a>

%H B. C. Berndt, A. Zaharescu, <a href="http://dx.doi.org/10.1007/s00208-004-0559-5">Finite trigonometric sums and class numbers</a>, Math. Ann. 330 (2004), 551-575.

%H B. C. Berndt, L.-C. Zhang, <a href="http://dx.doi.org/10.1007/BF01444636">Ramanujan's identities for eta-functions</a>, Math. Ann. 292 (1992), 561-573.

%H Z.-G. Liu, <a href="http://dx.doi.org/10.2140/pjm.2003.209.103">Some Eisenstein series identities related to modular equations of the seventh order</a>, Pacific J. Math. 209 (2003), 103-130.

%H Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.

%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7).

%F G.f.: (1-5*x+7*x^2)/(1-7*x+14*x^2-7*x^3).

%t LinearRecurrence[{7,-14,7},{1,2,7},40]

%o (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

%o (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

%Y Cf. A215007, A215008.

%K nonn,easy

%O 0,2

%A _Roman Witula_, Aug 04 2012