A259806 Number of n-step self-avoiding walks on the cubic lattice that start at the origin, stay in the top half-space, and end at height 0.

1, 4, 12, 40, 136, 528, 2032, 8344, 33576, 140912, 582088, 2482240, 10451064, 45101536, 192562328, 838630216, 3619073608, 15876846888, 69099916504, 304951643096

Offset

0,2

Comments

Such self-avoiding walks are sometimes called arches.

Guttmann-Torrie simple cubic lattice series coefficients c_n^{21}(Pi).

Links

Table of n, a(n) for n=0..19.

A. J. Guttmann and G. M. Torrie, Critical behaviour at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552.

P. K. Mishra, S. Kumar and Y. Singh, Force-induced desorption of a linear polymer chain adsorbed on an attractive surface, EPL, 69 (2005), 102; arXiv:cond-mat/0404191, 2004.

Example

For n < 3, all such walks are confined to the plane z=0, so a(n) = A001411(n).
For n = 3, four walks of the form "up, sideways, down" are added, so a(3) = A001411(3) + 4 = 40.

Crossrefs

Cf. A001411, A116903, A116904, A259807.

Sequence in context: A081875 A102433 A221652 * A100320 A064649 A149332

Adjacent sequences: A259803 A259804 A259805 * A259807 A259808 A259809

Keyword

nonn,more

Author

N. J. A. Sloane, Jul 06 2015

Extensions

New name, a(0) and a(16)-a(19) added by Andrei Zabolotskii, Apr 09 2026, from Mishra, Kumar and Singh's table of C_N(N_s, h) in three dimensions: a(n) = C_{n+3}(2, 0) = C_{n+2}(1, 1) = Sum_{N_s=2..n+1} C_{n+1}(N_s, 0).

Status

approved