|
|
A005708
|
|
a(n) = a(n-1) + a(n-6), with a(i) = 1 for i = 0..5.
(Formerly M0496)
|
|
33
|
|
|
1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17887, 22990, 29548, 37975, 48804, 62721, 80608, 103598, 133146, 171121, 219925, 282646
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
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>=6, a(n-6) = number of compositions of n in which each part is >=6. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 6. - Joerg Arndt, Jun 24 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=6, 2*a(n-6) equals the number of 2-colored compositions of n with all parts >=6, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+5) equals the number of binary words of length n having at least 5 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
Number of tilings of a 6 X n rectangle with 6 X 1 hexominoes. - M. Poyraz Torcuk, Mar 26 2022
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = term (1,1) in the 6 X 6 matrix [1,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1]; 1,0,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
For positive integers n and k such that k <= n <= 6*k and 5 divides n-k, define c(n,k) = binomial(k,(n-k)/5), and c(n,k)=0, otherwise. Then, for n>= 1, a(n) = sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([1/6-n/6, 1/3-n/6, 1/2-n/6, 2/3-n/6, 5/6-n/6, -n/6], [1/5-n/5, 2/5-n/5, 3/5- n/5, 4/5-n/5, -n/5], -6^6/5^5) for n>=25. - Peter Luschny, Sep 19 2014
|
|
MAPLE
|
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 5)}, unlabeled]: seq(count(SeqSetU, size=j), j=6..59); # Zerinvary Lajos, Oct 10 2006
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 5)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=5..58); # Zerinvary Lajos, Mar 26 2008
M := Matrix(6, (i, j)-> if j=1 and member(i, [1, 6]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1, 1]; seq(a(n), n=0..60); # Alois P. Heinz, Jul 27 2008
|
|
MATHEMATICA
|
LinearRecurrence[{1, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
|
|
PROG
|
(PARI) x='x+O('x^66); Vec(x/(1-(x+x^6))) /* Joerg Arndt, Jun 25 2011 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
|
|
STATUS
|
approved
|
|
|
|