[seqfan] Re: questions about walks in the plane

[John Machacek]

Hello,

For A001630 the walks are "ternary" words in {-1, +1, +3}, but the "length"
is sum of the absolute values (as opposed to length as a word in
{-1, +1, +3}).

So, a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) since

a(n-1) accounts for walks ending with (+1)
a(n-2) accounts for walks ending with (+1, -1)
a(n-3) accounts for walks ending with (+3)
a(n-4) accounts for walks ending with (+3, -1).

Then we check the initial conditions also work

a(2) = 0: (empty)
a(3) = 2: (+1), (-1)
a(4) = 3: (+1,+1), (+1,-1), (-1,+1)
a(5) = 6: (+1,+1,+1), (+1,+1,-1), (+1,-1,+1), (-1,+1,+1), (-1,+1,-1), (+3)

I don't currently have anything to say about the 2-D walks. But I'll try to
think about them...

Best,
John Machacek


On Sun, May 31, 2020 at 3:25 AM Nacin, David <NACIND at wpunj.edu> wrote:

> Though the last three questions are clear, I'm confused on the first.  If
> we are talking one-D walks using +1,-1, +3 with no consecutive -1's then
> there must be some other restriction as well, otherwise the sequence would
> just contain A028859<https://oeis.org/A028859>
> 1,3,8,22,60,164,448,1224,... .  (It also wouldn't matter what the numbers
> themselves were, only that one of the numbers can't be repeated
> consecutively.)  What am I missing?  Does the walk have to end at a certain
> value?
>
> -David
>
> ________________________________
> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Neil Sloane <
> njasloane at gmail.com>
> Sent: Thursday, May 28, 2020 12:26 PM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] questions about walks in the plane
>
> An old friend (Kees Immink) asked me about the conjecture of David Scambler
> in A085363.  In fact there are 4 assertions/conjectures in the OEIS of this
> type: (the first is only a one-D walk)
>
> %C A001630 Apparently for n>=2 the number of 1-D walks of length n-2 using
> steps +1, +3 and -1, avoiding consecutive -1 steps. - _David Scambler_, Jul
> 15 2013
>
> %C A084768 Number of directed 2-D walks of length 2n starting at (0,0) and
> ending on the X-axis using steps NE, SE, NE, SW and avoiding NE followed by
> SE. - _David Scambler_, Jun 24 2013
>
> %C A085363 Apparently, the number of 2-D directed walks of semilength n
> starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
> avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20 2013
>
> %C A101500 Directed 2-D walks with n steps starting at (0,0) and ending on
> the X-axis using steps N,S,E,W and avoiding N followed by S. - _David
> Scambler_, Jun 24 2013
>
> I know we have several experts here - could someone help and provide
> proofs?
>
> The third question is the following:
> Let a(n) = the number of 2-D directed walks of semilength n starting at
> (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding
> adjacent NW/SE and adjacent NE/SW
> Show that this satisfies the recurrence
> a(0)=1; for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
>
> (The second and fourth questions are stated as if they are theorems, but no
> proof is given.)
>
> Neil
>
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