0,2

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)

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

Also stacking bricks.

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

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

M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 360.

W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.

F. Faase, Counting Hamilton cycles in product graphs

F. Faase, Counting Hamilton cycles in product graphs

F. Faase, Results from the counting program

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.

Index entries for sequences related to dominoes

Index entries for sequences related to bricks

Nearest integer to (1/6)*(2+sqrt(3))^(n+1). - D. E. Knuth, Jul 15 1995.

For n >= 4, a(n) = 3a(n-1) + 3a(n-2) - a(n-3). - Avi Peretz (njk(AT)netvision.net.il), Mar 30 2001

For n >= 3, a(n) = 4a(n-1) - a(n-2) + 2*(-1)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 14 2001

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).

G.f.: (1-x)/((1+x)*(1-4*x+x^2)) = (1-x)/(1-3*x-3*x^2-x^3).

A006253:=-(-1+z)/(z+1)/(z**2-4*z+1); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]

Cf. A002530, A004003, A006125.

Sequence in context: A053369 A076959 A003697 this_sequence A045630 A075364 A026526

Adjacent sequences: A006250 A006251 A006252 this_sequence A006254 A006255 A006256

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).