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A005709
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a(n) = a(n-1) + a(n-7), with a(i) = 1 for i = 0..6.
(Formerly M0492)
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41
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1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851
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OFFSET
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0,8
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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_{i=0..n/m} binomial(n-(m-1)*i, i). 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 >= 7, a(n-7) is the number of compositions of n in which each part is >=7. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 7. - Joerg Arndt, Jun 24 2011
a(n+6) is the number of binary words of length n having at least 6 zeros between every two successive ones. - Milan Janjic, Feb 09 2015
Number of tilings of a 7 X n rectangle with 7 X 1 heptominoes. - M. Poyraz Torcuk, Feb 26 2022
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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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David Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
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FORMULA
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For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k)=0, otherwise. Then, for n >= 1, a(n) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([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], [1/6-n/6, 1/3-n/6, 1/2-n/6, 2/3-n/6, 5/6-n/6, -n/6], -7^7/6^6) for n >= 36. - Peter Luschny, Sep 19 2014
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MAPLE
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with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 6)}, unlabeled]: seq(count(SeqSetU, size=j), j=7..55); # Zerinvary Lajos, Oct 10 2006
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 6)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=6..54); # Zerinvary Lajos, Mar 26 2008
M:= Matrix(7, (i, j)-> if j=1 and member(i, [1, 7]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1, 1]; seq(a(n), n=0..50); # Alois P. Heinz, Jul 27 2008
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MATHEMATICA
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f[ n_Integer ] := f[ n ]=If[ n>7, f[ n-1 ]+f[ n-7 ], 1 ]
Table[Sum[Binomial[n-6*i, i], {i, 0, n/7}], {n, 0, 45}] (* Adi Dani, Jun 25 2011 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec(1/(1-(x+x^7))) /* Joerg Arndt, Jun 25 2011 */
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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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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