1,3

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.

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]

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. C. Fielder, Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217 (see Eq. 29).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..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.

G.f.: (1+3*x^2)/(1-x-x^3).

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

a(n)= A000930(n)+3*A000930(n-2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

A001609:=-(1+3*z**2)/(-1+z+z**3); [S. Plouffe in his 1992 dissertation.]

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

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

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

(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x^2)/(1-x-x^3)+x*O(x^n), n))

Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710.

Sequence in context: A066501 A114439 A079257 this_sequence A101590 A057916 A162415

Adjacent sequences: A001606 A001607 A001608 this_sequence A001610 A001611 A001612

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000. More terms from Michael Somos, Oct 03, 2002.