A005708 a(n) = a(n-1) + a(n-6), with a(i) = 1 for i = 0..5.

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

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>=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

• T. D. Noe, Table of n, a(n) for n=0..500
• Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
• 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 pp. 18, 22.
• 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.
• Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
• 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
• V. C. Harris and 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.
• Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
• S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4,5} gives seq. shifted 5 places left
• 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 [math.CO] (2016), Section 4.5
• 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.
• 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 379
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1).

Crossrefs

Cf. A000045, A000079, A000930, A003269, A003520, A005709, A005710, A005711.

Programs

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 *)

PARI

x='x+O('x^66); Vec(x/(1-(x+x^6))) /* Joerg Arndt, Jun 25 2011 */

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

Extensions

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