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A006603
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Generalized Fibonacci numbers.
(Formerly M1771)
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7
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1, 2, 7, 26, 107, 468, 2141, 10124, 49101, 242934, 1221427, 6222838, 32056215, 166690696, 873798681, 4612654808, 24499322137, 130830894666, 702037771647, 3783431872018, 20469182526595, 111133368084892, 605312629105205, 3306633429423460, 18111655081108453
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OFFSET
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0,2
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COMMENTS
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Typo on the right-hand side of Rogers's equation (1-x+x^2+x^3)*R^*(x) = R(x)+x: the sign in front of the x should be switched. - R. J. Mathar, Nov 23 2018
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1 - x - 2x^2 - sqrt(1 - 6x + x^2))/(2x*(1 - x + x^2 + x^3)) = (A006318(x) - x)/(1 - x + x^2 + x^3).
a(n) = Sum_{k=1..floor(n/2)+1} k*(1/(n-k+2))*Sum_{i=0..n-2*k+2} C(n-k+2,i)*C(2*n-3*k-i+3,n-k+1). - Vladimir Kruchinin, Oct 23 2011
(n+1)*a(n) +(-7*n+2)*a(n-1) +4*(2*n-1)*a(n-2) +6*(-n+1)*a(n-3) +(-5*n+1)*a(n-4) +(n-2)*a(n-5)=0. - R. J. Mathar, Nov 23 2018
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MAPLE
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A006603 := n-> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): seq(A006603(n), n=0..24); # Johannes W. Meijer, Jul 15 2013
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MATHEMATICA
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CoefficientList[Series[(1-x-2x^2-Sqrt[1-6x+x^2])/(2x(1-x+x^2+x^3)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 12 2016 *)
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PROG
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(Maxima) a(n):=sum((k*sum(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i, 0, n-2*k+2))/(n-k+2), k, 1, n/2+1); /* Vladimir Kruchinin, Oct 23 2011 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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