a[rule_, n_] := -Subtract @@ Flatten[Map[Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[rule], Prepend[Table[0, {n - 1}], 1], Unequal, All], {0}]] In[]= a[18, #] & /@ Range[32] Out[]= {1, 1, 1, 1, 2, 2, 1, 1, 6, 6, 4, 4, 14, 14, 1, 1, 14, 14, 12, 12, 62, 62, 8, 8, 126, 126, 28, 28, 30, 30, 1, 1} This one is also missing from OEIS, but I think it is just double A268754. Also, r18 = B1/S on a "wire" of height one? I think so, despite my confusion about the code, but for def. see: https://arxiv.org/pdf/0911.2890.pdf The sequence A268754 itself has "binary resonance", so falls under item 3 of my previous list. --Brad On Wed, May 6, 2020 at 12:45 PM Brad Klee <bradklee@gmail.com> wrote:
I double checked, and made a few more conjectures on case splitting:
1. Periodic cases: a(c1*(n+1)+c2)=a(c1*n+c2)+c3(c2) {190, 41, 9, 107, 25} -> {A334501, A334509, A334511, A33514, A33513}
2. For large n, a(n)=3 {62, 131} -> {A334502, A334503}
3. Sequence should be viewed as table with rows a(2^n),a(2^n+1),...,a(2^(n+1)-1) {26, 169, 161, 45, 105} -> {A334504, A334505, A334506, A334508, A334512} Also, 73->A334510, but the pattern is not as easy to see.
The most interesting case was anomalous 122 -> A334507. It is halfway periodic, only on the even terms:
In[]= a[122, 2 #] & /@ Range[1, 20] In[]=a[122, 2 # + 1] & /@ Range[1, 20]
Out[]= {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2} Out[]= {1, 2, 1, 6, 4, 14, 1, 14, 12, 62, 8, 126, 28, 30, 1, 30, 28, 1022, 24, 126}
Highly likely that the second sequence is http://oeis.org/A268754. Comparing with APG's mma def. Jan. 13 2009, definition via C.A. seems the more simple. This looks interesting, but I don't know the convention of A268754, so could not check rigorously just yet.
Maybe it is easy though. Adam, is there an isomorphism here?
--Brad
On Tue, May 5, 2020 at 3:56 PM Neil Sloane <njasloane@gmail.com> wrote:
Brad, Thanks for answering (on April 26) my questions about the period length of rule 30, etc.
I've just created sequences A334496, A334497, and A334501-A334515 as a result of your reply.