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%I A215493 #38 Sep 08 2022 08:46:03
%S A215493 0,1,4,14,49,175,637,2352,8771,32928,124166,469567,1779141,6749211,
%T A215493 25623472,97329337,369821228,1405502182,5342323441,20307982135,
%U A215493 77201862045,293497548512,1115812645899,4242135876440,16128056932078,61317184775679,233122447515741
%N A215493 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.
%C A215493 The Berndt-type sequence number 4 for the argument 2Pi/7 - see also A215007, A215008, A215143 and A215494.
%C A215493 We have a(n)=A079309(n) for n=1..6, and A079309(7)-a(7)=1.
%H A215493 G. C. Greubel, <a href="/A215493/b215493.txt">Table of n, a(n) for n = 0..1000</a>
%H A215493 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 A215493 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 A215493 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 A215493 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 A215493 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7).
%F A215493 a(n)*sqrt(7) = s(1)^(2n-1) + s(2)^(2n-1) + s(4)^(2n-1), where s(j) := 2*Sin(2*Pi*j/7) (for the sums of the respective even powers see A215494, see also A094429, A115146). For the proof of these formula see Witula-Slota's paper.
%F A215493 G.f.: x*(1-3*x)/(1-7*x+14*x^2-7*x^3).
%F A215493 a(n) = A275830(2*n-1)/(7^n). - _Kai Wang_, May 25 2017
%t A215493 LinearRecurrence[{7,-14,7}, {0,1,4}, 50]
%o A215493 (PARI) x='x+O('x^30); concat([0], Vec(x*(1-3*x)/(1-7*x+14*x^2-7*x^3))) \\ _G. C. Greubel_, Apr 23 2018
%o A215493 (Magma) I:=[0,1,4]; [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 A215493 nonn,easy
%O A215493 0,3
%A A215493 _Roman Witula_, Aug 13 2012

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