|
| |
|
|
A005708
|
|
a(n) = a(n-1) + a(n-6), with a(i) = 1 for i = 0..5.
(Formerly M0496)
|
|
26
|
|
|
|
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
|
|
|
REFERENCES
|
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=0..500
Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,5,1).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 (2016), Section 4.5
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 379
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1).
|
|
|
FORMULA
|
G.f.: 1/(1-x-x^6). - Simon Plouffe in his 1992 dissertation
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
|
Cf. A000045, A000079, A000930, A003269, A003520, A005709, A005710, A005711.
Sequence in context: A193286 A098132 A017900 * A085793 A281809 A143286
Adjacent sequences: A005705 A005706 A005707 * A005709 A005710 A005711
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
EXTENSIONS
|
Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
|
|
|
STATUS
|
approved
|
| |
|
|
