%I A005709 M0492
%S A005709 1,1,1,1,1,1,1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,53,66,83,
%T A005709 105,133,168,211,264,330,413,518,651,819,1030,1294,1624,2037,
%U A005709 2555,3206,4025,5055,6349,7973,10010,12565,15771,19796,24851
%N A005709 a(n)=a(n-1)+a(n-7).
%C A005709 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.
%D A005709 E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional 
               lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%D A005709 Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
%D A005709 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005709 T. D. Noe, <a href="b005709.txt">Table of n, a(n) for n=0..500</a>
%H A005709 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005709 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005709 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=380">
               Encyclopedia of Combinatorial Structures 380</a>
%p A005709 A005709 := proc(n) option remember; if n <=6 then 1; else A005709(n-1)+A005709(n-7); 
               fi; end;
%p A005709 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
%p A005709 A005709:=-1/(-1+z+z**7); [S. Plouffe in his 1992 dissertation.]
%p A005709 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
%p A005709 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
%t A005709 f[ n_Integer ] := f[ n ]=If[ n>7, f[ n-1 ]+f[ n-7 ], 1 ]
%Y A005709 Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005710, A005711.
%Y A005709 Sequence in context: A062010 A071218 A017901 this_sequence A101917 A127273 
               A143287
%Y A005709 Adjacent sequences: A005706 A005707 A005708 this_sequence A005710 A005711 
               A005712
%K A005709 nonn
%O A005709 0,8
%A A005709 N. J. A. Sloane (njas(AT)research.att.com).
%E A005709 Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000