A005710 a(n) = a(n-1) + a(n-8), with a(i) = 1 for i = 0..7.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• T. D. Noe, Table of n, a(n) for n = 0..500
• Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
• Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
• Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 18.
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 10.
• P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
• 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 and 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
• Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.
• D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
• A. O. Munagi, Euler-type identities for integer compositions via zig-zag graphs, Integers 12 (2012), Paper No. A60, 10 pp.
• Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
• D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
• Djamila Oudrar and Maurice Pouzet, Ordered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes, arXiv:2312.05913 [math.CO], 2023. See p. 18.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 381
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1).

Crossrefs

Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005709, A005711.

Programs

Maple

A005710:=-1/(-1+z+z**8); # Simon Plouffe in his 1992 dissertation.
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 *)

PARI

x='x+O('x^66); Vec(x/(1-(x+x^8))) /* Joerg Arndt, Jun 25 2011 */

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

Extensions

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000