This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033155 #37 Jul 08 2025 19:42:26
%S A033155 0,0,8,32,88,256,736,2032,5376,14224,36976,95504,243536,619168,
%T A033155 1559168,3916960,9769072,24321552,60199464,148803824,366051864,
%U A033155 899559584,2201636848,5384254000,13121348672,31957730688,77595810512
%N A033155 Configurations of linear chains for a square lattice.
%C A033155 From _Petros Hadjicostas_, Jan 03 2019: (Start)
%C A033155 In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence is C_{n,m} with m=1 (and d=2). Here, for a d-dimensional hypercubic lattice, C_{n,m} is "the number of configurations of an n-bond self-avoiding chain with m neighbor contacts."
%C A033155 These numbers are given in Table I (p. 1088) in the paper by Nemirovsky et al. (1992). Using Eqs. (5) and (7b) in the paper, we can prove that C_{n,m=1} = 2^1*1!*Bin(2,1)*p_{n,m=1}^{(1)} + 2^2*2!*Bin(2,2)*p_{n,m=1}^{(2)} = 0 + 8*p_{n,m=1}^{(2)} = 8*A038747(n).
%C A033155 (End)
%C A033155 The terms a(12) to a(21) were copied from Table B1 (pp. 4738-4739) in Bennett-Wood et al. (1998). In the table, the authors actually calculate a(n)/4 = C(n, m=1)/4 for 1 <= n <= 29. (They use the notation c_n(k), where k stands for m, which equals 1 here. They call c_n(k) "the number of SAWs of length n with k nearest-neighbour contacts".) - _Petros Hadjicostas_, Jan 04 2019
%H A033155 D. Bennett-Wood, I. G. Enting, D. S. Gaunt, A. J. Guttmann, J. L. Leask, A. L. Owczarek, and S. G. Whittington, <a href="https://doi.org/10.1088/0305-4470/31/20/010">Exact enumeration study of free energies of interacting polygons and walks in two dimensions</a>, J. Phys. A: Math. Gen. 31 (1998), 4725-4741.
%H A033155 M. E. Fisher and B. J. Hiley, <a href="http://dx.doi.org/10.1063/1.1731729">Configuration and free energy of a polymer molecule with solvent interaction</a>, J. Chem. Phys., 34 (1961), 1253-1267.
%H A033155 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a033/A033155.java">Java program</a> (github)
%H A033155 A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108; see Eq. 5 (p. 1090) and Eq. 7b (p. 1093).
%F A033155 a(n) = 8*A038747(n) for n >= 1. (It can be proved using Eqs. (5) and (7b) in the paper by Nemirovsky et al. (1992).) - _Petros Hadjicostas_, Jan 03 2019
%Y A033155 Cf. A038747.
%K A033155 nonn,more
%O A033155 1,3
%A A033155 _N. J. A. Sloane_
%E A033155 Name edited by _Petros Hadjicostas_, Jan 03 2019
%E A033155 a(22)-a(27) from _Sean A. Irvine_, Jul 03 2020