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 A334877 #27 Jul 14 2020 23:16:50 %S A334877 1,4,12,36,108,324,948,2740,7892,22540,64020,181396,511828,1440652, %T A334877 4045676,11322732,31615780,88100644,245143676,681002276,1888943100, %U A334877 5233741636,14484853148,40043579596,110590828396,305133547724 %N A334877 Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing length from 1 to n. %C A334877 This sequence gives the number of self-avoiding walks on a 2-dimensional square lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. %C A334877 The first time a collision with a previous step can occur is for n = 6. This can occur in three different ways. For example a walk with steps of length 1,2 and 3 to the right, a step of length 4 upward, then a step of length 5 to the left. A step of length 6 downward would now result in a collision. Requiring six steps before a collision is in contrast to the standard 2D square lattice SAW of A001411 where a collision can occur on the fourth step. %C A334877 Note that this sequence agrees with a SAW on the diamond lattice, A001394, for the first 7 terms, even though the seventh term here has some walks removed due to the above collision. %H A334877 A. R. Conway et al., <a href="http://dx.doi.org/10.1088/0305-4470/26/7/012">Algebraic techniques for enumerating self-avoiding walks on the square lattice</a>, J. Phys A 26 (1993) 1519-1534. %H A334877 A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/20/7/029">On the critical behavior of self-avoiding walks</a>, J. Phys. A 20 (1987), 1839-1854. %H A334877 A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345. %e A334877 a(1) = 4. These are the four directions one can step away from a point on a 2D square lattice. %e A334877 a(2) = 12. These consist of the two following walks: %e A334877 . %e A334877 * %e A334877 | 1 2 %e A334877 . 2 *---*---.---* %e A334877 | %e A334877 *---* %e A334877 1 %e A334877 . %e A334877 The first walk can be taken in 8 different ways, the second in 4 ways, giving a total of 12 walks. %e A334877 a(3) = 36. These consist of the following five walks: %e A334877 . %e A334877 * * %e A334877 | | %e A334877 . 3 3 . %e A334877 | 3 *---.---.---* *---.---.---* | 3 %e A334877 . | | . %e A334877 | . 2 . 2 | %e A334877 * | | *---*---.---* %e A334877 | *---* *---* 1 2 %e A334877 . 2 1 1 %e A334877 | *---*---.---*---.---.---* %e A334877 *---* 1 2 3 %e A334877 1 %e A334877 . %e A334877 The first four can be taken in 8 different ways, while the last straight walk can be taken in 4 ways, giving a total of 36 walks. Notice it is not possible to form a collision from any of these walks by adding a step of length 4. %Y A334877 Cf. A001411, A334720, A077482, A334720, A001394. %K A334877 nonn,more,walk %O A334877 0,2 %A A334877 _Scott R. Shannon_, May 13 2020