A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

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

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.

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

Number of tilings of a 5 X n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Mar 26 2022

References

• A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.
• 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.
• Roland Bacher, On the number of perfect lattices, arXiv:1704.02234 [math.NT], 2017. See Section 6.
• 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.
• 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,4,1).
• V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
• Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378
• S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4} gives seq. shifted 4 places left
• T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.
• 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 33.
• R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.4.
• 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), Article 16.8.3.
• 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
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1)

Crossrefs

Apart from initial terms, same as A017899.

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

Programs

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 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,e+a}; NestList[nxt,{1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Sep 27 2023 *)

Maxima

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

PARI

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

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

Extensions

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