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%I #39 Sep 08 2022 08:45:06
%S 1,2,7,6,21,18,63,54,189,162,567,486,1701,1458,5103,4374,15309,13122,
%T 45927,39366,137781,118098,413343,354294,1240029,1062882,3720087,
%U 3188646,11160261,9565938,33480783,28697814,100442349,86093442
%N Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = b*nu(n-1) + lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+...+q^(n-1)) and q is a root of unity.
%C Instead of listing the coefficients of the highest power of q in each nu(n), if we list the coefficients of the smallest power of q (i.e., constant terms), we get a sequence of weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n >= 2, f(n) = 2*f(n-1) + 3*f(n-2).
%H Vincenzo Librandi, <a href="/A072985/b072985.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Beattie, S. Dăscălescu and S. Raianu, <a href="https://arxiv.org/abs/math/0204075">Lifting of Nichols Algebras of Type B_2</a>, arXiv:math/0204075 [math.QA], 2002.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2) = b^2+lambda and for n >= 3, t(n) = lambda*t(n-2).
%F G.f.: (1 + 2*x + 4*x^2)/(1-3*x^2). - _R. J. Mathar_, Dec 05 2007
%F a(n) = 3*a(n-2) for n>2. - _Ralf Stephan_, Jul 19 2013
%F a(n) = (1/6)*(13 + (-1)^n)*3^floor(n/2) for n>0. - _Ralf Stephan_, Jul 19 2013
%e nu(0) = 1;
%e nu(1) = 2;
%e nu(2) = 7;
%e nu(3) = 20 + 6q;
%e nu(4) = 61 + 33q + 21q^2;
%e nu(5) = 182 + 144q + 120q^2 + 78q^3 + 18q^4;
%e nu(6) = 547 + 570q + 585q^2 + 501q^3 + 381q^4 + 162q^5 + 63q^6; ...
%e The coefficients of the highest power of q give this sequence.
%t CoefficientList[Series[-(1 + 2 x + 4 x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 20 2013 *)
%t Join[{1}, LinearRecurrence[{0, 3}, {2, 7}, 33]] (* _Jean-François Alcover_, Sep 23 2017 *)
%o (Magma) [1] cat [(1/6)*(13+(-1)^n)*3^Floor(n/2): n in [1..40]]; // _Vincenzo Librandi_, Jul 20 2013
%o (PARI) x='x+O('x^30); Vec((1+2*x+4*x^2)/(1-3*x^2)) \\ _G. C. Greubel_, May 26 2018
%Y Cf. A014983.
%K nonn,easy
%O 0,2
%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%E More terms from _R. J. Mathar_, Dec 05 2007
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