Search: id:A006253
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%I A006253 M1926
%S A006253 1,2,9,32,121,450,1681,6272,23409,87362,326041,1216800,4541161,16947842,
%T A006253 63250209,236052992,880961761,3287794050,12270214441,45793063712,170902040409,
%U A006253 637815097922,2380358351281,8883618307200,33154114877521,123732841202882
%N A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.
%C A006253 Number of tilings of a box with sides 2 X 2 X n in R^3 by boxes of sides 
               2 X 1 X 1 (3-dimensional dominoes). - Frans Faase (Frans_LiXia(AT)wxs.nl)
%C A006253 The number of domino tilings in A006253, A004003, A006125 is the number 
               of perfect matchings in the relevant graphs. There are results of 
               Jockusch and Ciucu that if a planar graph has a rotational symmetry 
               then the number of perfect matchings is a square or twice a square 
               - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com 
               or danfux(AT)OpenGaia.com), Apr 12 2001
%C A006253 Also stacking bricks.
%C A006253 a(n)*(-1)^n = (1-T(n+1,-2))/3, n>=0, with Chebyshev's polynomials T(n,
               x) of the first kind, is the r=-2 member of the r-family of sequences 
               S_r(n) defined in A092184 where more information can be found. W. 
               Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 
               2004
%D A006253 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006253 M. Ciucu, Enumeration of perfect matchings in graphs with reflective 
               symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
%D A006253 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 360.
%D A006253 W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory 
               Ser. A 67 (1994), no. 1, 100-115.
%H A006253 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting 
               Hamilton cycles in product graphs</a>
%H A006253 F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamilton 
               cycles in product graphs</a>
%H A006253 F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from 
               the counting program</a>
%H A006253 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 A006253 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 A006253 <a href="Sindx_Do.html#domino">Index entries for sequences related to 
               dominoes</a>
%H A006253 <a href="Sindx_Br.html#bricks">Index entries for sequences related to 
               bricks</a>
%F A006253 Nearest integer to (1/6)*(2+sqrt(3))^(n+1). - D. E. Knuth, Jul 15 1995.
%F A006253 For n >= 4, a(n) = 3a(n-1) + 3a(n-2) - a(n-3). - Avi Peretz (njk(AT)netvision.net.il), 
               Mar 30 2001
%F A006253 For n >= 3, a(n) = 4a(n-1) - a(n-2) + 2*(-1)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), 
               Apr 14 2001
%F A006253 Comment from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), 
               Apr 11 2001: The values are a(1) = 2 * 1^2, a(2) = 3^2, a(3) = 2 
               * 4^2, a(4) = 11^2, a(5) = 2 * 15^2, ... and in general for odd n 
               a(n) is twice a square, for even n a(n) is a square. If we define 
               b(n) by b(n) = sqrt(a(n)) for even n, b(n) = sqrt(a(n)/2) for odd 
               n then apart from the first 2 elements b(n) is A002530(n+1).
%F A006253 G.f.: (1-x)/((1+x)*(1-4*x+x^2)) = (1-x)/(1-3*x-3*x^2-x^3).
%p A006253 A006253:=-(-1+z)/(z+1)/(z**2-4*z+1); [Conjectured (correctly) by S. Plouffe 
               in his 1992 dissertation.]
%Y A006253 Cf. A002530, A004003, A006125.
%Y A006253 Sequence in context: A053369 A076959 A003697 this_sequence A045630 A075364 
               A026526
%Y A006253 Adjacent sequences: A006250 A006251 A006252 this_sequence A006254 A006255 
               A006256
%K A006253 nonn,easy
%O A006253 0,2
%A A006253 N. J. A. Sloane (njas(AT)research.att.com).

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