A175941 -id:A175941 - cp's OEIS frontend

A321535 Number of different ways a grasshopper can take n hops without landing on the same point more than once.

1, 1, 1, 1, 2, 3, 4, 7, 9, 14, 23, 35, 56, 86, 136, 216, 338, 535, 848, 1374, 2234, 3594, 5750, 9265, 14856, 24019, 39350, 64222, 104878, 170247, 276489, 452138, 739486, 1207429, 1974247, 3234889, 5295560, 8708262, 14276970, 23493811, 38683402, 63773042, 104840427

Offset: 0

Andrew Howroyd, Nov 12 2018

nonn

Consider a grasshopper (cf. A141000, A141002) that starts at x=0 at time 0, then makes successive hops of sizes 1, 2, 3, ..., n, subject to the constraints that it must always land on a point x >= 0 and no point may be visited more than once; sequence gives number of different ways that it can make n hops.

In other words, the number of n step self avoiding walks on a line where the n-th step has length n.

			a(6) = 4 because there are 4 walks with 6 steps:
0 -> 1 -> 3 -> 6 -> 2 -> 7 -> 13,
0 -> 1 -> 3 -> 6 -> 10 -> 5 -> 11,
0 -> 1 -> 3 -> 6 -> 10 -> 15 -> 9,
0 -> 1 -> 3 -> 6 -> 10 -> 15 -> 21.

Cf. A141000, A141002, A175941.

a(n)={local(f=vectorsmall(n*(n+1)/2+1)); my(recurse(p, k)=if(p>0&&!f[p], if(k==n, 1, f[p]=1; k++; my(z=self()(p+k, k) + self()(p-k, k)); f[p]=0; z))); recurse(1, 0)}

cp's OEIS Frontend

A321535 Number of different ways a grasshopper can take n hops without landing on the same point more than once.

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