Domino Tiling
The Fibonacci number 
gives the
number of ways for 
dominoes to cover a 
checkerboard,
as illustrated in the diagrams above (Dickau).
The numbers of domino tilings, also known as dimer coverings, of a 
square
for 
, 2, ... are given by 2, 36, 6728, 12988816, ...
(OEIS A004003). The 36 tilings on the 
square are illustrated above. A formula
for these numbers is given by
![A_n=2^(2n^2)product_(i=1)^nproduct_(j=1)^n[cos^2((ipi)/(2n+1))+cos^2((jpi)/(2n+1))].](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/DominoTiling/NumberedEquation1.gif) |
(1)
|
Writing
 |
(2)
|
gives the surprising result
 |
(3)
|
(John and Sachs 2000). For 
, 2, ..., the
first few terms are 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, ... (OEIS A143234).
Writing
(OEIS A143233), where 
is Catalan's
constant.
SEE ALSO: Domino,
Fibonacci
Number
REFERENCES:
Cohn, H. "2-adic Behavior of Numbers of Domino Tilings." Electronic
J. Combinatorics 6, No. 1, R14, 1-7, 1999. http://www.combinatorics.org/Volume_6/Abstracts/v6i1r14.html.
Dickau, R. M. "Fibonacci Numbers." http://www.prairienet.org/~pops/fibboard.html.
Finch, S. R. "Monomer-Dimer Constants." §5.23 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 406-407,
2003.
Fisher, M. E. "Statistical Mechanics of Dimers on a Plane Lattice."
Phys. Rev. 124, 1664-1672, 1961.
Jockusch, W. "Perfect Matchings and Perfect Squares." J. Combin. Theory
Ser. A 67, 100-115, 1994.
John, P. E. and Sachs, H. "On a Strange Observation in the Theory of the
Dimer Problem." Disc. Math. 216, 211-219, 2000.
Schroeppel, R. Item 111 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 48,
Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item111.
Propp, J. "A Reciprocity Theorem for Domino Tilings." Electronic J.
Combinatorics 8, No. 1, R18, 1-9, 2001. http://www.combinatorics.org/Volume_8/Abstracts/v8i1r18.html.
Sloane, N. J. A. Sequence A004003/M2160, A143233, and A143234
in "The On-Line Encyclopedia of Integer Sequences."
Referenced on Wolfram|Alpha:
Domino Tiling
CITE THIS AS:
Weisstein, Eric W. "Domino Tiling." From
MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DominoTiling.html
![1/(16pi^2)int_(-pi)^piint_(-pi)^piln[4+2costheta+2cosphi]dthetadphi](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/DominoTiling/Inline13.gif)
Contact the MathWorld Team
tiling