login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003269 a(n) = a(n-1) + a(n-4) with a(0) = 0, a(1) = a(2) = a(3) = 1.
(Formerly M0526)
76
0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 345, 476, 657, 907, 1252, 1728, 2385, 3292, 4544, 6272, 8657, 11949, 16493, 22765, 31422, 43371, 59864, 82629, 114051, 157422, 217286, 299915, 413966, 571388, 788674, 1088589, 1502555, 2073943 (list; graph; refs; listen; history; text; internal format)
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(binomial(n-(m-1)*i, i), i=0..n/m). 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

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

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

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

V. C. Harris, 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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 377

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

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.

E. Wilson, The Scales of Mt. Meru

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

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)); - 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

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, 40} ]

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

Table[Sum[Binomial[n - 3i - 1, i], {i, 0, 35}], {n, 0, 35}]

LinearRecurrence[{1, 0, 0, 1}, {0, 1, 1, 1}, 49] (* Robert G. Wilson v, Jul 12 2014 *)

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)

-- Reinhard Zumkeller, Feb 27 2011

CROSSREFS

Cf. A000045, A000079, A000930, A003520, A005708, A005709, A005710, A005711, A017898, A048718, A072827, A072850-A072856, A079955-A080014.

See A017898 for an essentially identical sequence.

Sequence in context: A099559 A247084 A017898 * A087221 A206739 A107586

Adjacent sequences:  A003266 A003267 A003268 * A003270 A003271 A003272

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Mohammad K. Azarian

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

Initial 0 prepended by N. J. A. Sloane, Apr 09 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified October 1 14:59 EDT 2016. Contains 276659 sequences.