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

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.

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.

T. D. Noe, Table of n, a(n) for n=0..500

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[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); [Conjectured by 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

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.

Adjacent sequences: A003517 A003518 A003519 this_sequence A003521 A003522 A003523

Sequence in context: A026483 A098131 A017899 this_sequence A101915 A022468 A050933

nonn,easy

njas

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

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