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A005711
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a(n) = a(n-1) + a(n-9) for n >= 9; a(n) = 1 for n=0..7; a(8) = 2.
(Formerly M0479)
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12
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1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64, 76, 91, 110, 134, 164, 201, 246, 300, 364, 440, 531, 641, 775, 939, 1140, 1386, 1686, 2050, 2490, 3021, 3662, 4437, 5376, 6516, 7902, 9588, 11638, 14128, 17149, 20811, 25248
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OFFSET
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0,9
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COMMENTS
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a(n+7) equals the number of binary words of length n having at least 8 zeros between every two successive ones. - Milan Janjic, Feb 09 2015
a(n) is the number of compositions of n+1 into parts 1 and 9. - Joerg Arndt, May 19 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^8)/(1-x-x^9).
For positive integers n and k such that k <= n <= 9*k, and 8 divides n-k, define c(n,k) = binomial(k,(n-k)/8), and c(n,k) = 0, otherwise. Then, for n>= 1, a(n-1) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
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MAPLE
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ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 8)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=9..65); # Zerinvary Lajos, Mar 26 2008
M:= Matrix(9, (i, j)-> if j=1 and member(i, [1, 9]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n+1))[1, 1]; seq(a(n), n=0..60); # Alois P. Heinz, Jul 27 2008
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MATHEMATICA
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CoefficientList[Series[(1+x^8)/(1-x-x^9), {x, 0, 57}], x] (* Michael De Vlieger, May 20 2018 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 2}, 60] (* Harvey P. Dale, Jul 30 2022 *)
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PROG
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(PARI) x='x+O('x^66); Vec((1+x^8)/(1-x-x^9)) /* Joerg Arndt, Jun 25 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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STATUS
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approved
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