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A005709
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a(n)=a(n-1)+a(n-7).
(Formerly M0492)
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17
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| 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>=7, a(n-7) = number of compositions of n in which each part is >=7. [From Milan R. Janjic (agnus(AT)blic.net), Jun 28 2010]
Number of compositions of n into parts 1 and 7. [Joerg Arndt, Jun 24 2011]
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REFERENCES
| E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 380
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FORMULA
| G.f.: 1/(1-x-x^7). [Joerg Arndt, Jun 24 2011]
For positive integers n and k such that k <= n <= 7*k, and 6 devides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k)=0, otherwise. Then, for n >= 1, a(n) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
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MAPLE
| A005709 := proc(n) option remember; if n <=6 then 1; else A005709(n-1)+A005709(n-7); fi; end;
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 6)}, unlabeled]: seq(count(SeqSetU, size=j), j=7..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 10 2006
A005709:=-1/(-1+z+z**7); [S. Plouffe in his 1992 dissertation.]
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 (zerinvarylajos(AT)yahoo.com), 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..48); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 27 2008
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MATHEMATICA
| 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] (* From Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
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PROG
| (PARI) x='x+O('x^66); Vec(1/(1-(x+x^7))) /* Joerg Arndt, Jun 25 2011 */
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CROSSREFS
| Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005710, A005711.
Sequence in context: A062010 A071218 A017901 * A101917 A127273 A143287
Adjacent sequences: A005706 A005707 A005708 * A005710 A005711 A005712
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KEYWORD
| nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
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