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%I #19 Sep 08 2022 08:46:03
%S 1,0,-2,-9,-33,-113,-376,-1235,-4032,-13126,-42673,-138641,-450293,
%T -1462292,-4748343,-15418256,-50063514,-162556377,-527819057,
%U -1713820537,-5564744720,-18068619435,-58668449392,-190495275070,-618534298433,-2008368291137,-6521130940157,-21173979252396,-68751478912175,-223234649986656,-724838355712626
%N a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=0, a(2)=-2.
%C The Berndt-type sequence number 10 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For additional informations and particularly connections with another sequences of trigonometric nature - see comments to A215512 (a(n) is equal to the sequence c(n) in these comments) and Witula-Slota's reference (Section 3).
%C The following summation formula hold true (see comments to A215512): Sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) + 1, n=3,4,...
%H G. C. Greubel, <a href="/A215695/b215695.txt">Table of n, a(n) for n = 0..1000</a>
%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_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,1).
%F sqrt(7)*a(n) = s(1)*c(1)^(2*n) + s(2)*c(2)^(2*n) + s(4)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
%F G.f.: (1-5*x+4*x^2)/(1-5*x+6*x^2-x^3).
%F a(n) = A005021(n) - 5*A005021(n-1) + 4*A005021(n-2). - _R. J. Mathar_, Aug 22 2012
%e We have a(8)=3*a(7)+3*a(5)-6*a(2) and a(9)=3*a(8)+3*a(6)-3*a(4)-a(1).
%t LinearRecurrence[{5,-6,1}, {1,0,-2}, 50]
%o (PARI) x='x+O('x^30); Vec((1-5*x+4*x^2)/(1-5*x+6*x^2-x^3)) \\ _G. C. Greubel_, Apr 25 2018
%o (Magma) I:=[1,0,-2]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Apr 25 2018
%Y Cf. A215512 (the inverse binomial transform, up to signs), A215694.
%K sign,easy
%O 0,3
%A _Roman Witula_, Aug 21 2012
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