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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

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.

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.

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

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Last modified November 19 11:04 EST 2017. Contains 294936 sequences.