A005709 - OEIS

A005709

a(n) = a(n-1) + a(n-7), with a(i) = 1 for i = 0..6.
(Formerly M0492)

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851

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OFFSET

0,8

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 >= 7, a(n-7) is the number of compositions of n in which each part is >=7. - Milan Janjic, Jun 28 2010

Number of compositions of n into parts 1 and 7. - Joerg Arndt, Jun 24 2011

a(n+6) is the number of binary words of length n having at least 6 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

Number of tilings of a 7 X n rectangle with 7 X 1 heptominoes. - M. Poyraz Torcuk, Feb 26 2022

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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 380

R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.6.

Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.

David 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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1).

FORMULA

G.f.: 1/(1-x-x^7). - Simon Plouffe in his 1992 dissertation.

For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), 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([1/7-n/7, 2/7-n/7, 3/7-n/7, 4/7-n/7, 5/7-n/7, 6/7-n/7, -n/7], [1/6-n/6, 1/3-n/6, 1/2-n/6, 2/3-n/6, 5/6-n/6, -n/6], -7^7/6^6) for n >= 36. - Peter Luschny, Sep 19 2014

MAPLE

A005709 := proc(n) option remember; if n <=6 then 1; else A005709(n-1)+A005709(n-7); fi; end;

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 6)}, unlabeled]: seq(count(SeqSetU, size=j), j=7..55); # Zerinvary Lajos, Oct 10 2006

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 6)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=6..54); # Zerinvary Lajos, Mar 26 2008

M:= Matrix(7, (i, j)-> if j=1 and member(i, [1, 7]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1, 1]; seq(a(n), n=0..50); # Alois P. Heinz, Jul 27 2008

MATHEMATICA

f[ n_Integer ] := f[ n ]=If[ n>7, f[ n-1 ]+f[ n-7 ], 1 ]

Table[Sum[Binomial[n-6*i, i], {i, 0, n/7}], {n, 0, 45}] (* Adi Dani, Jun 25 2011 *)

LinearRecurrence[{1, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

PROG

(PARI) x='x+O('x^66); Vec(1/(1-(x+x^7))) /* Joerg Arndt, Jun 25 2011 */

CROSSREFS

Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005710, A005711.

Sequence in context: A215775 A363149 A236310 * A017901 A101917 A322854

Adjacent sequences: A005706 A005707 A005708 * A005710 A005711 A005712

KEYWORD

nonn,easy