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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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.

Adjacent sequences: A003266 A003267 A003268 this_sequence A003270 A003271 A003272

Sequence in context: A017836 A099559 A017898 this_sequence A087221 A107586 A130080

nonn,new

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