0,6

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

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).

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

G.f.: 1/(1-x-x^5).

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

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

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[x/(1 - x - x^5), {x, 0, 51}], x] - Zerinvary Lajos (Zerinvary Lajos(zerinvarylajos(AT)yahoo.com), Mar 29 2007

Apart from initial terms, same as A017899.

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

Sequence in context: A026483 A098131 A017899 this_sequence A101915 A022468 A050933

Adjacent sequences: A003517 A003518 A003519 this_sequence A003521 A003522 A003523

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Mohammad K. Azarian (ma3(AT)evansville.edu)

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