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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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11
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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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Alois P. Heinz, Table of n, a(n) for n = 0..1000
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 382
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 1).
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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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A005711:=-(1+z**8)/(-1+z+z**9); # Simon Plouffe in his 1992 dissertation
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 *)
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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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Cf. A005710.
Sequence in context: A260768 A130224 A017903 * A322856 A280863 A059765
Adjacent sequences: A005708 A005709 A005710 * A005712 A005713 A005714
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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