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A003520
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a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.
(Formerly M0507)
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26
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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; internal format)
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OFFSET
| 0,6
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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 (Alford1940(AT)aol.com), 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
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REFERENCES
| A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
Index entries for sequences related to linear recurrences with constant coefficients
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378
E. Wilson, The Scales of Mt. Meru
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FORMULA
| G.f.: 1/(1-x-x^5).
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 (benoit7848c(AT)orange.fr), Nov 30 2002
a(n) = term (1,1) in the 5x5 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 (heinz(AT)hs-heilbronn.de), Jul 27 2008
For positive integers n and k such that k <= n <= 5*k, and 4 devides 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
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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 (zerinvarylajos(AT)yahoo.com), Oct 10 2006
A003520:=-1/(z**3+z**2-1)/(z**2-z+1); [S. 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 (zerinvarylajos(AT)yahoo.com), 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 (heinz(AT)hs-heilbronn.de), Jul 27 2008
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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}] (from Robert G. Wilson v Dec 09 2004)
CoefficientList[Series[1/(1 - x - x^5), {x, 0, 51}], x] - Zerinvary Lajos (Zerinvary Lajos(zerinvarylajos(AT)yahoo.com), Mar 29 2007
LinearRecurrence[{1, 0, 0, 0, 1}, {1, 1, 1, 1, 1}, 80] (* From Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
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PROG
| (Maxima)
a(n):=sum(binomial(n-1+(-4)*j, j), j, 0, (n-1)/4); [From Vladimir Kruchinin (kru(AT)ie.tusur.ru), May 23 2011]
(PARI) x='x+O('x^66); Vec(x/(1-(x+x^5))) /* Joerg Arndt, Jun 25 2011 */
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CROSSREFS
| Apart from initial terms, same as A017899.
Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.
Sequence in context: A026483 A098131 A017899 * A101915 A022468 A181324
Adjacent sequences: A003517 A003518 A003519 * A003521 A003522 A003523
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Mohammad K. Azarian (ma3(AT)evansville.edu)
Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
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