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A334877
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Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing length from 1 to n.
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11
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1, 4, 12, 36, 108, 324, 948, 2740, 7892, 22540, 64020, 181396, 511828, 1440652, 4045676, 11322732, 31615780, 88100644, 245143676, 681002276, 1888943100, 5233741636, 14484853148, 40043579596, 110590828396, 305133547724
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OFFSET
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0,2
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COMMENTS
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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.
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.
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.
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LINKS
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EXAMPLE
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a(1) = 4. These are the four directions one can step away from a point on a 2D square lattice.
a(2) = 12. These consist of the two following walks:
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*
| 1 2
. 2 *---*---.---*
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*---*
1
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The first walk can be taken in 8 different ways, the second in 4 ways, giving a total of 12 walks.
a(3) = 36. These consist of the following five walks:
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* *
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. 3 3 .
| 3 *---.---.---* *---.---.---* | 3
. | | .
| . 2 . 2 |
* | | *---*---.---*
| *---* *---* 1 2
. 2 1 1
| *---*---.---*---.---.---*
*---* 1 2 3
1
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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.
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CROSSREFS
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KEYWORD
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nonn,more,walk
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AUTHOR
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STATUS
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approved
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