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A074082
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Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).
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19
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0, 0, 0, 0, 2, 6, 16, 37, 81, 169, 342, 675, 1307, 2491, 4686, 8718, 16066, 29364, 53282, 96065, 172215, 307151, 545286, 963993, 1697701, 2979383, 5211852, 9090060, 15810530, 27429426, 47473828, 81983773, 141286221, 243011173
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OFFSET
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0,5
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COMMENTS
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The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).
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REFERENCES
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Paper in progress by Y. Kelly Itakura, to appear.
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LINKS
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Table of n, a(n) for n=0..33.
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols algebras of type B_2
Index to sequences with linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).
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FORMULA
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G.f.: (2x^4-2x^6-x^7)/(1-x-x^2)^3.
a(n)=3a(n-1)-5a(n-3)+3a(n-5)+a(n-6) for n>=8.
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EXAMPLE
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The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^2 are 0,0,0,0,2,6.
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MATHEMATICA
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b=1; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
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CROSSREFS
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Coefficients of q^0, q^1 and q^3 are in A000045, A023610 and A074083. Related sequences with different values of b and lambda are in A074084-A074089.
Sequence in context: A026540 A128232 A099099 * A212383 A097813 A167821
Adjacent sequences: A074079 A074080 A074081 * A074083 A074084 A074085
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KEYWORD
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nonn,easy
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AUTHOR
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Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Aug 21 2002
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STATUS
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approved
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