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%I #39 Sep 08 2022 08:46:03
%S 3,1,3,31,53,87,1011,1673,2771,32119,53189,88079,1020995,1690737,
%T 2799811,32454831,53744245,88998887,1031656755,1708393209,2829048851,
%U 32793751175,54305486341,89928286367,1042430160131,1726233651041,2858592097539,33136210400191
%N a(n) = 7^(floor(n/3))*A(n), where A(n) = A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, A(2)=3.
%C The Berndt-type sequence number 13 for the argument 2Pi/7
%C defined by the relation ((-sqrt(7))^n)*A(n) = t(1)^n + t(2)^n + t(4)^n = (-sqrt(7) + 4*s(1))^n + (-sqrt(7) + 4*s(2))^n + (-sqrt(7) + 4*s(4))^n, where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7), and the fact that all numbers 7^(floor(n/3))*A(n) are integers. We note that ((-sqrt(7))^n)*A(n) = B(n), where B(n) is defined in the comments to A215575. For more details see also A108716, A215794, Witula-Slota's (Section 6) and Witula's (Remark 11) papers.
%H Vincenzo Librandi, <a href="/A215828/b215828.txt">Table of n, a(n) for n = 0..1000</a>
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
%H R. Witula, P. Lorenc, M. Rozanski, 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 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_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,31,0,0,25,0,0,1).
%F G.f.: (x^8-5*x^7+25*x^6+6*x^5-22*x^4+62*x^3-3*x^2-x-3)/(x^9+25*x^6+31*x^3-1). [_Colin Barker_, Oct 28 2012]
%e We have A(3)=31/7, A(4)=53/7 and A(5)=87/7. On the other hand we have a(2)+a(3)+a(4)=a(5).
%t CoefficientList[Series[(x^8 - 5 x^7 + 25 x^6 + 6 x^5 - 22 x^4 + 62 x^3 - 3 x^2 - x - 3)/(x^9 + 25 x^6 + 31 x^3 - 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 19 2013 *)
%o (Magma) /* By definition: */ i:=28; I:=[3,1,3]; A:=[m le 3 select I[m] else Self(m-1)+Self(m-2)+Self(m-3)/7: m in [1..i]]; [7^(Floor((n-1)/3))*A[n]: n in [1..i]]; // _Bruno Berselli_, Oct 28 2012
%Y Cf. A215575, A108716, A215794.
%K nonn,easy
%O 0,1
%A _Roman Witula_, Aug 24 2012
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