%I A001609 M3240 N1308
%S A001609 1,1,4,5,6,10,15,21,31,46,67,98,144,211,309,453,664,973,1426,2090,3063,
%T A001609 4489,6579,9642,14131,20710,30352,44483,65193,95545,140028,205221,300766,
%U A001609 440794,646015,946781,1387575,2033590,2980371,4367946,6401536,9381907
%N A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).
%C A001609 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 = 1...m-1, 
               a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also 
               a(n) = 1 + n*sum(binomial(n-1-(m-1)*i, i-1)/i, i=1..n/m). This gives 
               the number of ways to cover (without overlapping) a ring lattice 
               (or necklace) of n sites with molecules that are m sites wide. Special 
               cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: 
               A058367, m=7: A058366, m=8: A058365, m=9: A058364.
%C A001609 The sequence defined by a(n)-1 plays a role for the computation of A065414, 
               A146486, A146487, and A146488 equivalent to the role of A001610 for 
               A005596, A146482, A146483 and A146484, see the variable a_{2,n} in 
               arXiv:0903.2514. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 28 2009]
%D A001609 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 A001609 D. C. Fielder, Special integer sequences controlled by three parameters. 
               Fibonacci Quart 6 1968 64-70.
%D A001609 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A001609 M. Newman and D. Shanks, On a sequence arising in series for pi, Math. 
               Comp., 42 (1984), 199-217 (see Eq. 29).
%D A001609 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001609 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001609 T. D. Noe, <a href="b001609.txt">Table of n, a(n) for n=1..500</a>
%H A001609 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 A001609 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.
%F A001609 G.f.: (1+3*x^2)/(1-x-x^3).
%F A001609 a(n) = trace of successive powers of matrix{{{0,0,1},{1,0,0},{0,1,1}})^n 
               - Artur Jasinski (grafix(AT)csl.pl), Jan 10 2007
%F A001609 a(n)= A000930(n)+3*A000930(n-2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 16 2007
%p A001609 A001609:=-(1+3*z**2)/(-1+z+z**3); [S. Plouffe in his 1992 dissertation.]
%t A001609 Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] 
               - Artur Jasinski (grafix(AT)csl.pl), Jan 10 2007
%t A001609 Table[ HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -(n/3)}, {1/2 - n/2, 
               1 - n/2}, -(27/4)], {n, 20}] - Alexander Povolotsky, Nov 21 2008
%t A001609 a[1] = a[2] = 1; a[3] = 4; m = 3; a[n_] := 1 + n*Sum [Binomial [n - 1 
               - (m - 1)*i, i - 1]/i, {i, n/m}] A001609 = Table[a[n], {n, 100}] 
               - Zak Seidov, Nov 21 2008
%o A001609 (PARI) a(n)=if(n<0,0,polcoeff((1+3*x^2)/(1-x-x^3)+x*O(x^n),n))
%Y A001609 Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710.
%Y A001609 Sequence in context: A066501 A114439 A079257 this_sequence A101590 A057916 
               A162415
%Y A001609 Adjacent sequences: A001606 A001607 A001608 this_sequence A001610 A001611 
               A001612
%K A001609 nonn
%O A001609 1,3
%A A001609 N. J. A. Sloane (njas(AT)research.att.com).
%E A001609 Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000. 
               More terms from Michael Somos, Oct 03, 2002.