A175941 Number of n-step self-avoiding walks on a line, where step X skips X - 1 spaces.

1, 2, 4, 6, 10, 18, 30, 50, 78, 130, 210, 350, 586, 954, 1606, 2588, 4234, 6944, 11342, 18948, 31450, 52206, 85662, 141680, 233040, 385428, 644910, 1072074, 1783342, 2953094, 4897922, 8157096, 13571014, 22552212, 37486916, 62325564, 103508754, 172765524, 287428656, 479052200, 798944976

Offset: 0

Author

Grant Garcia, Oct 25 2010

Keywords

• nonn
• walk

Examples

			a(0) = 1 because every walk must start at a specific point on the line.
   --0--
a(1) = 2 because there are two ways to start, moving one space forward or backward.
   --01-
   -10--
a(2) = 4 because there are two ways to continue each of the walks described in a(1).
   ---01-2
   --201--
   --102--
   2-10---
a(3) = 6, not 8, because two of the possible four continuations are not self-avoiding.
   ------01-2--3
   -----2013----
   --3--201-----
   -----102--3--
   ----3102-----
   3--2-10------
		

Crossrefs

Cf. A321535.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=0, 1, add(`if`(j in s, 0,
      b(n-1, {map(x-> x-j, s)[], 0})), j=(k->[-k, k])(nops(s))))
    end:
a:= n-> b(n, {0}):
seq(a(n), n=0..24);  # Alois P. Heinz, May 19 2021

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, 1, Sum[If[MemberQ[s, j], 0, b[n - 1, Union[Append[s - j, 0]]]], {j, {-Length[s], Length[s]}}]];
a[n_] := b[n, {0}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 09 2021, after Alois P. Heinz *)

PARI

a(n)={local(f=vectorsmall(n*(n+1)+1)); my(recurse(p, k)=if(!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((#f+1)/2, 0)} \\ Andrew Howroyd, Nov 12 2018

Python

x = 1
n = 0
runs = [[n]]
while x < 30:
    print(x - 1, n)
    runs2 = []
    n = 0
    while runs:
        for pmx in (x, -x):
            if runs[0][ -1] + pmx not in runs[0]:
                runs2.append(runs[0] + [runs[0][ -1] + pmx])
                n += 1
        del runs[0]
    runs = runs2
    x += 1

Formula

a(n) <= 2*a(n-1) for n > 0. - Andrew Howroyd, Nov 12 2018

Extensions

a(29)-a(40) from Andrew Howroyd, Nov 12 2018