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A005710
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a(n) = a(n-1) + a(n-8), with a(i) = 1 for i = 0..7.
(Formerly M0483)
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23
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1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623
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OFFSET
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0,9
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COMMENTS
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This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = sum(binomial(n-(m-1)*i, i), i=0..n/m). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
For n>=8, a(n-8) = number of compositions of n in which each part is >=8. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 8. - Joerg Arndt, Jun 24 2011
A005710 a(n+7) equals the number of binary words of length n having at least 7 zeros between every two successive ones. - Milan Janjic, Feb 09 2015
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REFERENCES
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P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194.
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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T. D. Noe, Table of n, a(n) for n=0..500
D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 10
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
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
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
A. O. Munagi, Euler-type identities for integer compositions via zig-zag graphs, Integers 12 (2012), Paper No. A60, 10 pp.
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 and Conjectures, Université du Québec à Montréal, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 381
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 1).
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FORMULA
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G.f.: 1/(1-x-x^8).
For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k)= 0, otherwise. Then, for n >= 1, a(n-1) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([1/8-n/8,1/4-n/8,3/8-n/8,1/2-n/8, 5/8-n/8, 3/4-n/8,7/8-n/8,-n/8], [1/7-n/7, 2/7-n/7, 3/7-n/7, 4/7-n/7, 5/7-n/7, 6/7-n/7, -n/7], -8^8/7^7) for n>=49. - Peter Luschny, Sep 19 2014
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MAPLE
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A005710:=-1/(-1+z+z**8); # Simon Plouffe in his 1992 dissertation.
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 7)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=7..62); # Zerinvary Lajos, Mar 26 2008
M := Matrix(8, (i, j)-> if j=1 and member(i, [1, 8]) then 1 elif (i=j-1) then 1 else 0 fi); a := n -> (M^(n))[1, 1]; seq (a(n), n=0..55); # Alois P. Heinz, Jul 27 2008
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
CoefficientList[Series[1/(1-x-x^8), {x, 0, 60}], x] (* Harvey P. Dale, Jun 14 2016 *)
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PROG
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(PARI) x='x+O('x^66); Vec(x/(1-(x+x^8))) /* Joerg Arndt, Jun 25 2011 */
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CROSSREFS
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Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005709, A005711.
Sequence in context: A079064 A123176 A017902 * A291146 A023358 A322855
Adjacent sequences: A005707 A005708 A005709 * A005711 A005712 A005713
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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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EXTENSIONS
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Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
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STATUS
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approved
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