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

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.

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.

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.

F. Faase, Counting Hamilton cycles in product graphs

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 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

njas