|
| |
|
|
A001609
|
|
a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).
(Formerly M3240 N1308)
|
|
11
| |
|
|
1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
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 = 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]
|
|
|
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.
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).
Z. Skupien, Sparse Hamiltonian 2-decompositions ..., Discr. Math., 309 (2009), 6382-6390.
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).
|
|
|
LINKS
| 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.
|
|
|
FORMULA
| 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
|
|
|
MAPLE
| A001609:=-(1+3*z**2)/(-1+z+z**3); [S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| 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
|
|
|
PROG
| (PARI) a(n)=if(n<0, 0, polcoeff((1+3*x^2)/(1-x-x^3)+x*O(x^n), n))
|
|
|
CROSSREFS
| Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710.
Sequence in context: A186808 A114439 A079257 * A101590 A057916 A162415
Adjacent sequences: A001606 A001607 A001608 * A001610 A001611 A001612
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000. More terms from Michael Somos, Oct 03, 2002.
|
| |
|
|
