A003269 - OEIS

OFFSET

0,6

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 this family of sequences, a(n+1) is the number of compositions of n+1 into parts 1 and m. For n>=m, a(n-m+1)is the number of compositions of n in which each part is greater than m or equivalently, in which parts 1 through m are excluded. - Gregory L. Simay, Jul 14 2016

For this family of sequences, let a(m,n) = a(n-1) + a(n-m). Then the number of compositions of n having m as a least summand is a(m, n-m) - a(m+1, n-m-1). - Gregory L. Simay, Jul 14 2016

For n>=3, a(n-3) = number of compositions of n in which each part is >=4. - Milan Janjic, Jun 28 2010

For n>=1, number of compositions of n into parts == 1 (mod 4). Example: a(8)=5 because there are 5 compositions of 8 into parts 1 or 5: (1,1,1,1,1,1,1,1), (1,1,1,5), (1,1,5,1), (1,5,1,1), (5,1,1,1). - Adi Dani, Jun 16 2011

a(n+1) is the number of compositions of n into parts 1 and 4. - Joerg Arndt, Jun 25 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>=4, 2*a(n-3) equals the number of 2-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=3, I={1,2}. - Vladimir Baltic, Mar 07 2012

a(n+4) equals the number of binary words of length n having at least 3 zeros between every two successive ones. - Milan Janjic, Feb 07 2015

From Clark Kimberling, Jun 13 2016: (Start)

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*.

Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3, 2*x, x+1, x^2}, etc.

Let T(r) be the tree obtained by substituting r for x.

If N is a positive integer such that r = N^(1/4) is not an integer, then the number of (not necessarily distinct) integers in g(n) is A003269(n), for n > = 1. See A274142. (End)

REFERENCES

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 120.

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..501

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.

Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21. With 2 more initial zeros.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135.

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 9.

Bruce M. Boman, Geometric Capitulum Patterns Based on Fibonacci p-Proportions, Fibonacci Quart. 58 (2020), no. 5, 91-102.

Russ Chamberlain, Sam Ginsburg and Chi Zhang, Generating Functions and Wilf-equivalence on Theta_k-embeddings, University of Wisconsin, April 2012.

P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]

Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, Schreier Multisets and the s-step Fibonacci Sequences, arXiv:2304.05409 [math.CO], 2023. See also Integers (2024) Vol. 24A, Art. No. A7, p. 3.

Adi Dani, Compositions of natural numbers over arithmetic progressions

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

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012.

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,3,1).

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.

Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 377

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.

R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 17.

R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO] (2014), eq. (23).

Flaviano Morone, Ian Leifer, and Hernán A. Makse, Fibration symmetries uncover the building blocks of biological networks, Proceedings of the National Academy of Sciences (2020) Vol. 117, No. 15, 8306-8314.

Augustine 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.

Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.8.3.

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

E. Wilson, The Scales of Mt. Meru

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.

Index entries for two-way infinite sequences

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

FORMULA

G.f.: x/(1-x-x^4).

G.f.: -1 + 1/(1-Sum_{k>=0} x^(4*k+1)).

a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) for n>4.

a(n) = floor(d*c^n + 1/2) where c is the positive real root of -x^4+x^3+1 and d is the positive real root of 283*x^4-18*x^2-8*x-1 (c=1.38027756909761411... and d=0.3966506381592033124...). - Benoit Cloitre, Nov 30 2002

Equivalently, a(n) = floor(c^(n+3)/(c^4+3) + 1/2) with c as defined above (see A086106). - Greg Dresden and Shuer Jiang, Aug 31 2019

a(n) = term (1,2) in the 4 X 4 matrix [1,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,0]^n. - Alois P. Heinz, Jul 27 2008

From Paul Barry, Oct 20 2009: (Start)

a(n+1) = Sum_{k=0..n} C((n+3*k)/4,k)*((1+(-1)^(n-k))/2 + cos(Pi*n/2))/2;

a(n+1) = Sum_{k=0..n} C(k,floor((n-k)/3))(2*cos(2*Pi*(n-k)/3)+1)/3. (End)

a(n) = Sum_{j=0..(n-1)/3} binomial(n-1-3*j,j) (cf. A180184). - Vladimir Kruchinin, May 23 2011

A017817(n) = a(-4 - n) * (-1)^n. - Michael Somos, Jul 12 2003

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + x^3)/( x*(2*k+2 + x^3) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013

Appears a(n) = hypergeometric([1/4-n/4,1/2-n/4,3/4-n/4,1-n/4], [1/3-n/3,2/3-n/3,1-n/3], -4^4/3^3) for n>=10. - Peter Luschny, Sep 18 2014

EXAMPLE

G.f.: x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...

The number of compositions of 12 having 4 as a least summand is a(4, 12 -4 + 1) - a(5, 12 - 5 + 1) = A003269(9) - A003520(8) = 7-4 = 3. The compositions are (84), (48) and (444). - Gregory L. Simay, Jul 14 2016

MAPLE

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 3)}, unlabeled]: seq(count(SeqSetU, size=j), j=4..51);

seq(add(binomial(n-3*k, k), k=0..floor(n/3)), n=0..47); # Zerinvary Lajos, Apr 03 2007

A003269:=z/(1-z-z**4); # Simon Plouffe in his 1992 dissertation

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 3)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=3..50); # Zerinvary Lajos, Mar 26 2008

M:= Matrix(4, (i, j)-> if j=1 then [1, 0, 0, 1][i] elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1, 2]; seq(a(n), n=0..48); # Alois P. Heinz, Jul 27 2008

MATHEMATICA

a[0]= 0; a[1]= a[2]= a[3]= 1; a[n_]:= a[n]= a[n-1] + a[n-4]; Table[a[n], {n, 0, 50}]

CoefficientList[Series[x/(1-x-x^4), {x, 0, 50}], x] (* Zerinvary Lajos, Mar 29 2007 *)

Table[Sum[Binomial[n-3*i-1, i], {i, 0, (n-1)/3}], {n, 0, 50}]

LinearRecurrence[{1, 0, 0, 1}, {0, 1, 1, 1}, 50] (* Robert G. Wilson v, Jul 12 2014 *)

nxt[{a_, b_, c_, d_}]:={b, c, d, a+d}; NestList[nxt, {0, 1, 1, 1}, 50][[;; , 1]] (* Harvey P. Dale, May 27 2024 *)

PROG

(PARI) {a(n) = polcoeff( if( n<0, (1 + x^3) / (1 + x^3 - x^4), 1 / (1 - x - x^4)) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Jul 12 2003 */

(Haskell)

a003269 n = a003269_list !! n

a003269_list = 0 : 1 : 1 : 1 : zipWith (+) a003269_list

(drop 3 a003269_list)