Search: id:A001224
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%I A001224 M0318 N0117
%S A001224 1,2,2,4,5,9,12,21,30,51,76,127,195,322,504,826,1309,2135,3410,5545,
%T A001224 8900,14445,23256,37701,60813,98514,159094,257608,416325,673933,
%U A001224 1089648,1763581,2852242,4615823,7466468,12082291,19546175,31628466
%N A001224 Packing a box with n dominoes.
%C A001224 Slavik V. Jablan (jablans(AT)yahoo.com) observes that this is also number 
               of generating rational knots and links. See reference.
%C A001224 Also the number of distinct binding configurations on an n-site one-dimensional 
               linear lattice, where the molecules cannot touch each other. This 
               number determines the order of recurrence for the partition function 
               of binding to a two-dimensional n x m lattice.
%D A001224 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001224 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001224 Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World 
               Scientific Press, 2007.
%D A001224 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.
%D A001224 S. Golomb, Polyominoes, Princeton Univ. Press 1994.
%D A001224 Y. Kong, General recurrence theory of ligand binding on a three-dimensional 
               lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799. (Table I)
%D A001224 W. E. Patton, Problem E1470, Amer. Math. Monthly, 69 (1962), 61-62.
%H A001224 T. D. Noe, <a href="b001224.txt">Table of n, a(n) for n=1..500</a>
%H A001224 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 A001224 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 A001224 C. G. Bower, <a href="transforms2.html">Transforms (2)</a>
%H A001224 S. V. Jablan, <a href="http://www.ams.org./preprints/57/199706/199706-57-001-199706-57-001.html">
               Geometry of Links</a>, XII Yugoslav Geometric Seminar (Novi Sad, 
               1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.
%H A001224 <a href="Sindx_Do.html#domino">Index entries for sequences related to 
               dominoes</a>
%F A001224 G.f.: (2-(x+x^2)^2)/(2*(1-x-x^2)) + (1+x+x^2)*(x^2+x^4)/(2*(1-x^2-x^4)).
%F A001224 "BIK" transform of x+x^2.
%F A001224 If F(n) is the n-th Fibonacci number, then a(2n)=(F(2n+1)+F(n+2))/2 and 
               a(2n+1)=(F(2n+2)+F(n+1))/2.
%p A001224 A001224:=-(-1-z+2*z**2+z**3+z**4+z**5)/(z**4+z**2-1)/(z**2+z-1); [Conjectured 
               by S. Plouffe in his 1992 dissertation.]
%p A001224 a:= n-> (Matrix([[5,4,2,2,1,1]]). Matrix(6, (i,j)-> if (i=j-1) then 1 
               elif j=1 then [1,2,-1,0,-1,-1][i] else 0 fi)^n)[1,6]: seq (a(n), 
               n=1..38); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 26 
               2008]
%Y A001224 Essentially the same as A060312.
%Y A001224 Sequence in context: A038000 A124280 A088518 this_sequence A102526 A050192 
               A007147
%Y A001224 Adjacent sequences: A001221 A001222 A001223 this_sequence A001225 A001226 
               A001227
%K A001224 nonn,nice,easy
%O A001224 1,2
%A A001224 N. J. A. Sloane (njas(AT)research.att.com).
%E A001224 More terms and formula from Christian G. Bower (bowerc(AT)usa.net), May 
               09 2000

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