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 A074323 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)=(1,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. 7
 1, 1, 3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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)+2f(n-2). The highest powers are given by the quarter-squares A002620(n-1). - Paul Barry, Mar 11 2007 LINKS M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002. Index entries for linear recurrences with constant coefficients, signature (0,2). FORMULA 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). G.f.: (1+x+x^2)/(1-2*x^2); a(n)=2^floor(n/2)+2^((n-2)/2)*(1+(-1)^n)/2-0^n/2. - Paul Barry, Mar 11 2007 a(0)=0, a(2n+1) = A000079, a(2n+2) = 3a(2n+1). a(2n)-a(2n+1) = A131577. - Paul Curtz, Mar 05 2008 a(2n+1) = 2^n = A000079(n), a(2n+2) = 3*A000079(n). Also a(2n)-a(2n+1) = A131577. a(2n+1)-a(2n)=2^n for n>0. - Paul Curtz, Apr 09 2008 a(n+1) = A010684(n)*A016116(n). - R. J. Mathar, Jul 08 2009 EXAMPLE nu(0)=1; nu(1)=1; nu(2)=3; nu(3)=5+2q; nu(4)=11+8q+6q^2; nu(5)=21+22q+20q^2+14q^3+4q^4; nu(6)=43+60q+70q^2+64q^3+54q^4+28q^5+12q^6; by listing the coefficients of the highest power in each nu(n), we get 1,1,3,2,6,4,12,... MATHEMATICA Join[{1}, LinearRecurrence[{0, 2}, {1, 3}, 41]] (* Jean-François Alcover, Sep 22 2017 *) CROSSREFS Cf. A001045. Sequence in context: A092401 A222208 A116626 * A162255 A164073 A286595 Adjacent sequences:  A074320 A074321 A074322 * A074324 A074325 A074326 KEYWORD nonn,easy,changed AUTHOR Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002 EXTENSIONS More terms from Paul Barry, Mar 11 2007 STATUS approved

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