Search: id:A003520
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%I A003520 M0507
%S A003520 1,1,1,1,1,2,3,4,5,6,8,11,15,20,26,34,45,60,80,106,140,185,245,325,431,
               571,756,
%T A003520 1001,1326,1757,2328,3084,4085,5411,7168,9496,12580,16665,22076,29244,
%U A003520 38740,51320,67985,90061,119305,158045,209365,277350,367411,486716,644761
%N A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.
%C A003520 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.
%C A003520 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
%D A003520 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 119.
%D A003520 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 A003520 T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money 
               management, Fib. Quart., 14 (1976), 37-41.
%D A003520 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003520 T. D. Noe, <a href="b003520.txt">Table of n, a(n) for n=0..500</a>
%H A003520 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A003520 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 A003520 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 A003520 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=378">
               Encyclopedia of Combinatorial Structures 378</a>
%H A003520 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales 
               of Mt. Meru</a>
%F A003520 G.f.: 1/(1-x-x^5).
%F A003520 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
%F A003520 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
%p A003520 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);
%p A003520 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
%p A003520 A003520:=-1/(z**3+z**2-1)/(z**2-z+1); [S. Plouffe in his 1992 dissertation.]
%p A003520 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
%p A003520 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
%t A003520 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)
%t A003520 CoefficientList[Series[x/(1 - x - x^5), {x, 0, 51}], x] - Zerinvary Lajos 
               (Zerinvary Lajos(zerinvarylajos(AT)yahoo.com), Mar 29 2007
%Y A003520 Apart from initial terms, same as A017899.
%Y A003520 Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.
%Y A003520 Sequence in context: A026483 A098131 A017899 this_sequence A101915 A022468 
               A050933
%Y A003520 Adjacent sequences: A003517 A003518 A003519 this_sequence A003521 A003522 
               A003523
%K A003520 nonn,easy
%O A003520 0,6
%A A003520 N. J. A. Sloane (njas(AT)research.att.com).
%E A003520 More terms from Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A003520 Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

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