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Search: id:A074323
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| A074323 |
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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. |
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+0
8
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| 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
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OFFSET
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0,3
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COMMENT
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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)=f(n-1)+2f(n-2).
The highest powers are given by the quarter-squares A002620(n-1). - Paul Barry (pbarry(AT)wit.ie), Mar 11 2007
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LINKS
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M. Beattie, S. D\u{a}sc\u{a}lescu and S. Raianu, Lifting of Nichols Algebras of Type $B_2$
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FORMULA
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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)/(1-2x^2); a(n)=2^floor(n/2)+2^((n-2)/2)*(1+(-1)^n)/2-0^n/2; - Paul Barry (pbarry(AT)wit.ie), Mar 11 2007
a(0)=0, a(2n+1)=A000079, a(2n+2)=3a(2n+1). a(2n)-a(2n+1) = A131577. - Paul Curtz (bpcrtz(AT)free.fr), Mar 05 2008
a(2n+1)=2^n=A000079(n), a(2n+2)=3*A000079(n). Also a(2n)-a(2n+1)=A131577. For b(n)=0 before a(n) : b(2n+1)-b(2n)=2^n. - Paul Curtz (bpcrtz(AT)free.fr), Apr 09 2008
a(n+1)=A010684(n)*A016116(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 08 2009]
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EXAMPLE
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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,...
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CROSSREFS
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Cf. A001045.
Sequence in context: A092401 A116626 A162255 this_sequence A164073 A090571 A088452
Adjacent sequences: A074320 A074321 A074322 this_sequence A074324 A074325 A074326
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KEYWORD
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nonn
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AUTHOR
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Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
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EXTENSIONS
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More terms from Paul Barry (pbarry(AT)wit.ie), Mar 11 2007
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