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A215007 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=1, a(1)=3, a(2)=9. 23


%S 1,3,9,28,91,308,1078,3871,14161,52479,196196,737793,2785160,10540390,

%T 39955041,151615947,575723785,2187128524,8311078307,31587815308,

%U 120069510526,456434707519,1735184512425,6596692255391,25079305566420

%N a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=1, a(1)=3, a(2)=9.

%C The sequence a(n) we shall call Berndt-type sequence of number 1 for the argument 2*Pi/7 - our motivation from Berndt's et al. and myself papers comes (see the first formula below, which is in agreement with the respective identities discussed in these papers).

%C 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 R. Witula, E. Hetmaniok, 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 G. C. Greubel, <a href="/A215007/b215007.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 R. Wituła, P. Lorenc, M. Różański, 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.

%F 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 G.f.: (1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3).

%p 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 LinearRecurrence[{7,-14,7}, {1,3,9}, 30] (* _G. C. Greubel_, Feb 01 2018 *)

%o (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 (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 (Sage)

%o def A215007_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3)).list()

%o A215007_list(30) # _G. C. Greubel_, Oct 03 2019

%o (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 Cf. A122068, A215008.

%K nonn,easy

%O 0,2

%A _Roman Witula_, Jul 31 2012

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Last modified July 8 01:25 EDT 2020. Contains 335502 sequences. (Running on oeis4.)