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.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 120.

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

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

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..501

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 377

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.

E. Wilson, The Scales of Mt. Meru

a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) for n>4.

a(n) = floor(d*c^n + 1/2) where c is the positive real root of -x^4+x^3+1 and d is the positive real root of 283*x^4-18*x^2-8*x-1 ( c=1.38027756909761411... and d=0.3966506381592033124...) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 30 2002

G.f.: x/(1-x-x^4).

a(n) = term (1,2) in the 4x4 matrix [1,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,0]^n. - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 27 2008

Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 20 2009: (Start)

a(n+1)=sum{k=0..n, C((n+3k)/4,k)*((1+(-1)^(n-k))/2+cos(pi*n/2))/2};

a(n+1)=sum{k=0..n, C(k,floor((n-k)/3))(2*cos(2*pi*(n-k)/3)+1)/3}. (End)

G.f.: x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 +etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 3)}, unlabeled]: seq(count(SeqSetU, size=j), j=4..51);

seq(add(binomial(n-3*k, k), k=0..floor(n/3)), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007

A003269:=z/(1-z-z**4); [S. Plouffe in his 1992 dissertation.]

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 3)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=3..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2008

M := Matrix(4, (i, j)-> if j=1 then [1, 0, 0, 1][i] elif (i=j-1) then 1 else 0 fi); a := n -> (M^(n))[1, 2]; seq (a(n), n=0..48); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 27 2008

a[0] = 0; a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[n - 1] + a[n - 4]; Table[ a[n], {n, 0, 40} ]

CoefficientList[Series[x/(1 - x - x^4), {x, 0, 50}], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2007

(PARI) a(n)=polcoeff(if(n<0, (1+x^3)/(1+x^3-x^4), 1/(1-x-x^4))+x*O(x^abs(n)), abs(n))

(Other) sage: taylor( mul(x/(1 - x - x^4) for i in xrange(1, 2)), x, 0, 48)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]

Cf. A000045, A000079, A000930, A003520, A005708, A005709, A005710, A005711, A017898, A048718.

See A017898 for an essentially identical sequence.

A017817(n)=a(-4-n)(-1)^n.

Sequence in context: A017836 A099559 A017898 this_sequence A087221 A107586 A130080

Adjacent sequences: A003266 A003267 A003268 this_sequence A003270 A003271 A003272

nonn

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

Initial 0 prepended by N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2008