Search: id:A003269
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%I A003269 M0526
%S A003269 0,1,1,1,1,2,3,4,5,7,10,14,19,26,36,50,69,95,131,181,250,345,476,657,907,
               1252,1728,
%T A003269 2385,3292,4544,6272,8657,11949,16493,22765,31422,43371,59864,82629,
%U A003269 114051,157422,217286,299915,413966,571388,788674,1088589,1502555,2073943
%N A003269 a(n)=a(n-1)+a(n-4); a(0)=0, a(1)=a(2)=a(3)=1.
%C A003269 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.
%D A003269 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 120.
%D A003269 E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional 
               lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%D A003269 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, 
               Math. Ann., 45 (1894), 371-380.
%D A003269 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003269 T. D. Noe, <a href="b003269.txt">Table of n, a(n) for n=0..501</a>
%H A003269 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A003269 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A003269 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=377">
               Encyclopedia of Combinatorial Structures 377</a>
%H A003269 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A003269 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A003269 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales 
               of Mt. Meru</a>
%F A003269 a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) for n>4.
%F A003269 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
%F A003269 G.f.: x/(1-x-x^4).
%F A003269 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
%F A003269 Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 20 2009: (Start)
%F A003269 a(n+1)=sum{k=0..n, C((n+3k)/4,k)*((1+(-1)^(n-k))/2+cos(pi*n/2))/2};
%F A003269 a(n+1)=sum{k=0..n, C(k,floor((n-k)/3))(2*cos(2*pi*(n-k)/3)+1)/3}. (End)
%e A003269 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]
%p A003269 with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 3)}, 
               unlabeled]: seq(count(SeqSetU, size=j), j=4..51);
%p A003269 seq(add(binomial(n-3*k,k),k=0..floor(n/3)),n=0..47); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 03 2007
%p A003269 A003269:=z/(1-z-z**4); [S. Plouffe in his 1992 dissertation.]
%p A003269 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
%p A003269 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
%t A003269 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} ]
%t A003269 CoefficientList[Series[x/(1 - x - x^4), {x, 0, 50}], x] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Mar 29 2007
%o A003269 (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))
%o A003269 (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]
%Y A003269 Cf. A000045, A000079, A000930, A003520, A005708, A005709, A005710, A005711, 
               A017898, A048718.
%Y A003269 See A017898 for an essentially identical sequence.
%Y A003269 A017817(n)=a(-4-n)(-1)^n.
%Y A003269 Sequence in context: A017836 A099559 A017898 this_sequence A087221 A107586 
               A130080
%Y A003269 Adjacent sequences: A003266 A003267 A003268 this_sequence A003270 A003271 
               A003272
%K A003269 nonn
%O A003269 0,6
%A A003269 N. J. A. Sloane (njas(AT)research.att.com).
%E A003269 More terms from Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A003269 Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
%E A003269 Initial 0 prepended by N. J. A. Sloane (njas(AT)research.att.com), Apr 
               09 2008

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