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%I #30 Sep 08 2022 08:46:03
%S 0,7,35,147,588,2303,8918,34300,131369,501809,1913597,7289436,
%T 27748357,105581574,401620072,1527436967,5808448779,22086364419,
%U 83978326796,319298327159,1213996265902,4615645568660,17548659548105,66719552736809,253665154464813
%N a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=7, a(2)=35.
%C The Berndt-type sequence number 6 for the argument 2Pi/7 (see A215007, A215008, A215143, A215493 and A215494 for the respective sequences numbers 1-5) is defined by the following relation: a(n) = s(1)*s(2)^(2n+1) + s(2)*s(4)^(2n+1) + s(4)*s(1)^(2n+1), where s(j) := 2*sin(2*Pi*j/7). For the respective sums with even powers see A215143.
%C We note that a(4)=49*sqrt(7)*(s(1)*s(4)^(-6) + s(2)*s(4)^(-6) + s(4)*s(1)^(-6)) - see the respective value of the sequence y*(n) in Witula-Slota's paper.
%H G. C. Greubel, <a href="/A215510/b215510.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 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 Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7).
%F G.f.: (7*x-14*x^2)/(1-7*x+14*x^2-7*x^3).
%F a(n) = 7*A215008(n). - _R. J. Mathar_, Nov 07 2015
%e We have (1-7*x+14*x^2-7*x^3)*(a(1)*x + a(3)*x^2 + a(5)*x^3 + ...) = b(1)*x - b(2)*x^2 + b(3)*x^3 - b(4)*x^4 + (b(5)-2b(2))*x^5 + ..., where b(n)=A094430(n) for n=1,...,5.
%t LinearRecurrence[{7,-14,7}, {0,7,35}, 50]
%o (PARI) x='x+O('x^30); concat([0], Vec((7*x-14*x^2)/(1-7*x+14*x^2-7*x^3))) \\ _G. C. Greubel_, Apr 23 2018
%o (Magma) I:=[0,7,35]; [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 23 2018
%K nonn,easy
%O 0,2
%A _Roman Witula_, Aug 14 2012
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