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A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.
(Formerly M0507)
43
1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350, 367411, 486716, 644761 (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.

Also counts ordered partitions such that no part is less than 5. For example, a(12) = a(11) + a(7) where a(7) counts 11,6+5 and 5+6 and a(11) counts 15,10+5, 9+6,8+7,7+8,6+9,5+10 and 5+5+5. Thus a(12) = 3 + 8 = 11. a(12) counts 16,11+5,10+6,9+7,8+8,7+9,6+10 and 6+5+5 but also 5+11,5+6+5 and 5+5+6. Similar results hold for the other sequences formed by a(n) = a(n-1) + a(n-k). - Alford Arnold, Aug 06 2003

Number of compositions of n into parts 1 and 5. - 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>=5, 2*a(n-5) equals the number of 2-colored compositions of n with all parts >=5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

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

REFERENCES

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.

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

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

Roland Bacher, On the number of perfect lattices, arXiv:1704.02234 [math.NT], 2017. See Section 6.

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

R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988

V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,1).

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378

T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.

R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 33.

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), Article 16.8.3.

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 linear recurrences with constant coefficients, signature (1,0,0,0,1)

FORMULA

G.f.: 1/(1-x-x^5) = 1/((1-x+x^2)(1-x^2-x^3)).

a(n) = sum(j=0..(n-1)/4, binomial(n-1+(-4)*j,j)).

For n>5, a(n) = floor( d*c^n + 1/2) where c is the positive real root of x^5-x^4-1 and d is the positive real root of 161*x^3-23*x^2-12*x-1 ( c=1.32471795724474602... and d=0.3811571478326847...) - Benoit Cloitre, Nov 30 2002

a(n) = term (1,1) in the 5 X 5 matrix [1,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008

For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k)=0, otherwise. Then, for n >= 1,  a(n) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

Apparently a(n) = hypergeometric([-1/5*n, 1/5-1/5*n, 2/5-1/5*n, 3/5-1/5*n, 4/5-1/5*n], [-1/4*n, 1/4-1/4*n, 1/2-1/4*n, 3/4-1/4*n], -5^5/4^4) for n>=16. - Peter Luschny, Sep 18 2014

7*a(n) = A117373(n+4) +5*b(n) +4*b(n-1) +b(n-2) where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017

MAPLE

a[0]:=1:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=1:for n from 5 to 60 do a[n]:=a[n-1]+a[n-5] od:seq(a[n], n=0..60);

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 4)}, unlabeled]: seq(count(SeqSetU, size=j), j=5..55); # Zerinvary Lajos, Oct 10 2006

A003520:=-1/(z**3+z**2-1)/(z**2-z+1); # Simon Plouffe in his 1992 dissertation

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 4)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=4..54); # Zerinvary Lajos, Mar 26 2008

M := Matrix(5, (i, j)-> if j=1 then [1, 0, 0, 0, 1][i] 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

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[n - 1] + a[n - 5]; Table[ a[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 09 2004 *)

CoefficientList[Series[1/(1 - x - x^5), {x, 0, 51}], x] (* Zerinvary Lajos, Mar 29 2007 *)

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

PROG

(Maxima) a(n):=sum(binomial(n-1+(-4)*j, j), j, 0, (n-1)/4); /* Vladimir Kruchinin, May 23 2011 */

(PARI) x='x+O('x^66); Vec(x/(1-(x+x^5))) /* Joerg Arndt, Jun 25 2011 */

CROSSREFS

Apart from initial terms, same as A017899.

Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.

Sequence in context: A026483 A098131 A017899 * A101915 A295073 A282504

Adjacent sequences:  A003517 A003518 A003519 * A003521 A003522 A003523

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

STATUS

approved

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Last modified November 20 04:05 EST 2017. Contains 294959 sequences.