%I #41 Sep 08 2022 08:45:07

%S 1,1,4,3,12,9,36,27,108,81,324,243,972,729,2916,2187,8748,6561,26244,

%T 19683,78732,59049,236196,177147,708588,531441,2125764,1594323,

%U 6377292,4782969,19131876,14348907,57395628,43046721,172186884,129140163

%N a(2n+1) = 3^n, a(2n) = 4*3^(n-1) except for a(0) = 1.

%C Also: 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)=(1,3), 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 weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=f(n-1)+3f(n-2).

%C Sequences A162766, A166552 are essentially the same. - _M. F. Hasler_, Dec 03 2014

%H Vincenzo Librandi, <a href="/A074324/b074324.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+x+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)*(7+(-1)^n)*3^floor(n/2) for n>0. - _Ralf Stephan_, Jul 19 2013

%e nu(0)=1;

%e nu(1)=1;

%e nu(2)=4;

%e nu(3)=7+3q;

%e nu(4)=19+15q+12q^2;

%e nu(5)=40+45q+42q^2+30q^3+9q^4;

%e nu(6)=97+147q+180q^2+168q^3+147q^4+81q^5+36q^6;

%e by listing the coefficients of the highest power in each nu(n), we get, 1,1,4,3,12,9,36,....

%t CoefficientList[Series[-(1 + x + x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 20 2013 *)

%t LinearRecurrence[{0,3},{1,1,4},40] (* _Harvey P. Dale_, Mar 13 2016 *)

%o (Magma) [1] cat [(1/6)*(7+(-1)^n)*3^Floor(n/2):n in [1..40]]; // _Vincenzo Librandi_, Jul 20 2013

%o (PARI) a(n)=3^(n\2)\(3/4)^!bittest(n,0) \\ _M. F. Hasler_, Dec 03 2014

%Y Cf. A006130.

%K nonn,easy

%O 0,3

%A Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

%E More terms from _R. J. Mathar_, Dec 05 2007

%E Simpler definition from _M. F. Hasler_, Dec 03 2014