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%I A215007 #41 Jul 03 2023 20:20:50
%S A215007 1,3,9,28,91,308,1078,3871,14161,52479,196196,737793,2785160,10540390,
%T A215007 39955041,151615947,575723785,2187128524,8311078307,31587815308,
%U A215007 120069510526,456434707519,1735184512425,6596692255391,25079305566420
%N A215007 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=1, a(1)=3, a(2)=9.
%C A215007 The sequence {a(n)} we shall call the Berndt-type sequence of type 1 for the argument 2*Pi/7; our motivation comes from Berndt's et al. and my papers (see the first formula below, which is in agreement with the respective identities discussed in these papers).
%C A215007 We note that a(n) = A105849(n) for n=0,1,...,5, and A105849(6) - a(6) = 1. Moreover we have a(n) = 2*A215008(n) - A215008(n+1).
%D A215007 R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
%H A215007 G. C. Greubel, <a href="/A215007/b215007.txt">Table of n, a(n) for n = 0..1000</a>
%H A215007 B. C. Berndt and 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 A215007 B. C. Berndt and 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 A215007 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 A215007 Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H A215007 R. Wituła, P. Lorenc, M. Różański, and M. Szweda, <a href="http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-142e1ad8-f56c-4c6e-b6e6-f71d1f31e76e">Sums of the rational powers of roots of cubic polynomials</a>, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
%H A215007 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, -14, 7).
%F A215007 a(n) = (1/sqrt(7))*(cot(8*Pi/7)*(s(1))^2n + cot(4*Pi/7)*(s(4))^2n + cot(2*Pi/7)*(s(2))^2n), where s(j) := 2*sin(2Pi*j/7).
%F A215007 G.f.: (1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3).
%p A215007 seq(coeff(series((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 03 2019
%t A215007 LinearRecurrence[{7,-14,7}, {1,3,9}, 30] (* _G. C. Greubel_, Feb 01 2018 *)
%o A215007 (PARI) Vec((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3)+O(x^30)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o A215007 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3))) // _G. C. Greubel_, Feb 01 2018
%o A215007 (Sage)
%o A215007 def A215007_list(prec):
%o A215007     P.<x> = PowerSeriesRing(ZZ, prec)
%o A215007     return P((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3)).list()
%o A215007 A215007_list(30) # _G. C. Greubel_, Oct 03 2019
%o A215007 (GAP) a:=[1,3,9];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # _G. C. Greubel_, Oct 03 2019
%Y A215007 Cf. A122068, A215008.
%K A215007 nonn,easy
%O A215007 0,2
%A A215007 _Roman Witula_, Jul 31 2012

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