|
| |
|
|
A005710
|
|
a(n) = a(n-1) + a(n-8), with a(i) = 1 for i = 0..7.
(Formerly M0483)
|
|
31
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,9
|
|
|
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 = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..n/m} binomial(n-(m-1)*i, i). 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.
For n >= 8, a(n-8) = number of compositions of n in which each part is >= 8. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 8. - Joerg Arndt, Jun 24 2011
a(n+7) equals the number of binary words of length n having at least 7 zeros between every two successive ones. - Milan Janjic, Feb 09 2015
|
|
|
REFERENCES
|
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: 1/(1-x-x^8).
For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k) = 0, otherwise. Then, for n >= 1, a(n-1) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([1/8-n/8, 1/4-n/8, 3/8-n/8, 1/2-n/8, 5/8-n/8, 3/4-n/8, 7/8-n/8, -n/8], [1/7-n/7, 2/7-n/7, 3/7-n/7, 4/7-n/7, 5/7-n/7, 6/7-n/7, -n/7], -8^8/7^7) for n >= 49. - Peter Luschny, Sep 19 2014
|
|
|
MAPLE
|
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 7)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=7..62); # Zerinvary Lajos, Mar 26 2008
M := Matrix(8, (i, j)-> if j=1 and member(i, [1, 8]) then 1 elif (i=j-1) then 1 else 0 fi); a := n -> (M^(n))[1, 1]; seq(a(n), n=0..55); # Alois P. Heinz, Jul 27 2008
|
|
|
MATHEMATICA
|
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
CoefficientList[Series[1/(1-x-x^8), {x, 0, 60}], x] (* Harvey P. Dale, Jun 14 2016 *)
|
|
|
PROG
|
(PARI) x='x+O('x^66); Vec(x/(1-(x+x^8))) /* Joerg Arndt, Jun 25 2011 */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,changed
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
|
|
|
STATUS
|
approved
|
| |
|
|
