cp's OEIS Frontend

A072946 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)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

%I A072946 #21 Oct 06 2016 14:17:14
%S A072946 1,2,6,4,12,8,24,16,48,32,96,64,192,128,384,256,768,512,1536,1024,
%T A072946 3072,2048,6144,4096,12288,8192,24576,16384,49152,32768,98304,65536,
%U A072946 196608,131072,393216,262144,786432,524288,1572864,1048576,3145728,2097152
%N A072946 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)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.
%C A072946 Instead of listing the coefficients of the highest power of q in each nu(n), if we listed 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)=2f(n-1)+2f(n-2).
%H A072946 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 A072946 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2).
%F A072946 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 A072946 O.g.f.: (1+2*x+4*x^2)/(1-2*x^2). - _R. J. Mathar_, Dec 05 2007
%e A072946 nu(0)=1,
%e A072946 nu(1)=2,
%e A072946 nu(2)=6,
%e A072946 nu(3)=16+4q,
%e A072946 nu(4)=44+20q+12q^2,
%e A072946 nu(5)=120+80q+64q^2+40q^3+8q^4,
%e A072946 nu(6)=328+288q+280q^2+232q^3+168q^4+64q^5+24q^6.
%e A072946 By listing the coefficients of the highest power in each nu(n) we get 1,2,6,4,12,8,24,...
%t A072946 LinearRecurrence[{0,2},{1,2,6},50] (* _Harvey P. Dale_, Dec 31 2015 *)
%Y A072946 Essentially the same as A162255 and A164073.
%Y A072946 Cf. A002605.
%K A072946 nonn,easy
%O A072946 0,2
%A A072946 Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%E A072946 More terms from _R. J. Mathar_, Dec 05 2007