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A001635
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A Fielder sequence: a(n) = a(n-1) + a(n-2) - a(n-6), n >= 7.
(Formerly M0762 N0289)
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1
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0, 2, 3, 6, 10, 11, 21, 30, 48, 72, 110, 171, 260, 401, 613, 942, 1445, 2216, 3401, 5216, 8004, 12278, 18837, 28899, 44335, 68018, 104349, 160089, 245601, 376791, 578057, 886830, 1360538, 2087279, 3202216, 4912704, 7536863, 11562737, 17739062, 27214520
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OFFSET
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1,2
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COMMENTS
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This is an application of the general formula that Paul Barry gives for sequence A000129 to the subsequence of odd-indexed terms. - Pat Costello (pat.costello(AT)eku.edu), May 20 2003
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.: x^2*(2 + x + x^2 + x^3 - 5*x^4)/(1 - x - x^2 + x^6).
a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5), n >= 6.
a(n) = Sum_{k=0..n} C(2*n+1, 2*k+1) * 2^k. - Pat Costello (pat.costello(AT)eku.edu), May 20 2003
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MAPLE
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A001635:=-z*(2+3*z+4*z**2+5*z**3)/(-1+z**2+z**3+z**4+z**5); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
a := n -> (Matrix([[5, -1$3, 3, 4]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [1$2, 0$3, -1][i] else 0 fi)^n)[1, 1] ; seq (a(n), n=1..39); # Alois P. Heinz, Aug 01 2008
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MATHEMATICA
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LinearRecurrence[{1, 1, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 11}, 50] (* T. D. Noe, Aug 09 2012 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff(x^2*(2+x+x^2+x^3-5*x^4)/(1-x-x^2+x^6)+x*O(x^n), n))
(Magma) I:=[0, 2, 3, 6, 10, 11]; [n le 6 select I[n] else Self(n-1) + Self(n-2) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jan 09 2018
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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