Geodesic
Close packing of rectilinear polymers on a~square lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 39 (1979) no. 3, pp. 347-352

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The set of close packings of rectilinear $r$-mers on a square lattice is considered. It is shown that the number of configurations of $r$-reefs on a lattice containing $N$ sites increases with increasing $N$ not slower than $\exp{\{4GN/\pi r^2\} }$ and not faster than $(r/2)^{N/r^2}\exp{\{4GN/\pi r^2\} }$ if $r$ is even and $$ \biggl(\frac{r-1}{2}\biggr)^{N/r^2} \exp\biggl\{(N/\pi r^2)\int_0^{\pi} \operatorname{arch}\biggl(\frac{2r}{r-1}-\cos{\varphi}\biggr)\,d\varphi\biggr\}, $$ if $r$ is odd ($G$ is Catalan's constant). 
@article{TMF_1979_39_3_a5,
     author = {N. D. Gagunashvili and V. B. Priezzhev},
     title = {Close packing of rectilinear polymers on a~square lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {347--352},
     publisher = {mathdoc},
     volume = {39},
     number = {3},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1979_39_3_a5/}
}
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N. D. Gagunashvili; V. B. Priezzhev. Close packing of rectilinear polymers on a~square lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 39 (1979) no. 3, pp. 347-352. http://geodesic.mathdoc.fr/item/TMF_1979_39_3_a5/