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.

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.

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).

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); [Conjectured by S. Plouffe in his 1992 dissertation.]

Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] (*Artur Jasinski*) - Artur Jasinski (grafix(AT)csl.pl), Jan 10 2007

(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.

Adjacent sequences: A001606 A001607 A001608 this_sequence A001610 A001611 A001612

Sequence in context: A066501 A114439 A079257 this_sequence A101590 A057916 A007606

nonn

njas

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