This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A160239 #200 Jan 14 2024 07:19:16
%S A160239 1,8,8,24,8,64,24,112,8,64,64,192,24,192,112,416,8,64,64,192,64,512,
%T A160239 192,896,24,192,192,576,112,896,416,1728,8,64,64,192,64,512,192,896,
%U A160239 64,512,512,1536,192,1536,896,3328,24,192,192,576,192,1536,576,2688,112,896,896,2688,416,3328,1728,6784
%N A160239 Number of "ON" cells in a 2-dimensional cellular automaton ("Fredkin's Replicator") evolving according to the rule that a cell is ON in a given generation if and only if there was an odd number of ON cells among the eight nearest neighbors in the preceding generation, starting with one ON cell.
%C A160239 This is the odd-rule cellular automaton defined by OddRule 757 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - _N. J. A. Sloane_, Feb 25 2015
%C A160239 The partial sums are in A245542, in which the structure also looks like an irregular stepped pyramid. - _Omar E. Pol_, Jan 29 2015
%H A160239 N. J. A. Sloane, <a href="/A160239/b160239.txt">Table of n, a(n) for n = 0..10000</a>
%H A160239 Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.
%H A160239 Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.
%H A160239 Charles R Greathouse IV, <a href="/A160239/a160239.pdf">Compact illustration of generations 0..17</a>
%H A160239 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A160239 N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_1.pdf">Enlarged illustration of first 16 generations (pdf).</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239.png">Illustration for a(15) = 416 (png).</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_2.pdf">Illustration for a(15) = 416 (pdf).</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_1.png">Illustration for a(31) = 1728.</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_4.png">Illustration for a(63) = 6784.</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_1.tiff">Illustration for a(127) = 27392 (tiff).</a>
%H A160239 N. J. A. Sloane, <a href="/A160239/a160239_6.png">Illustration for a(127) = 27392 (png).</a>
%H A160239 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A160239 Alexander Yu. Vlasov, <a href="https://arxiv.org/abs/2312.13034">Modelling reliability of reversible circuits with 2D second-order cellular automata</a>, arXiv:2312.13034 [nlin.CG], 2023. See page 13.
%H A160239 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A160239 a(0) = 1; a(2t)=a(t), a(4t+1)=8*a(t), a(4t+3)=2*a(2t+1)+8*a(t) for t >= 0. (Conjectured by _Hrothgar_, Jul 11 2014; proved by _N. J. A. Sloane_, Oct 04 2014.)
%F A160239 For n >= 2, a(n) = 8^r * Product_{lengths i of runs of 1 in binary expansion of n} R(i), where r is the number of runs of 1 in the binary expansion of n and R(i) = A083424(i-1) = (5*4^(i-1)+(-2)^(i-1))/6. Note that row i of the table in A245562 lists the lengths of runs of 1 in binary expansion of i. Example: n=7 = 111 in binary, so r=1, i=3, R(3) = A083424(2) = 14, and so a(7) = 8^1*14 = 112. That is, this sequence is the Run Length Transform of A246030. - _N. J. A. Sloane_, Oct 04 2014
%F A160239 The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - _N. J. A. Sloane_, Aug 25 2014
%e A160239 From _Omar E. Pol_, Jul 22 2014 (Start):
%e A160239 Written as an irregular triangle in which row lengths is A011782 the sequence begins:
%e A160239 1;
%e A160239 8;
%e A160239 8, 24;
%e A160239 8, 64, 24, 112;
%e A160239 8, 64, 64, 192, 24, 192, 112, 416;
%e A160239 8, 64, 64, 192, 64, 512, 192, 896, 24, 192, 192, 576, 112, 896, 416, 1728;
%e A160239 8, 64, 64, 192, 64, 512, 192, 896, 64, 512, 512, 1536, 192, 1536, 896, 3328, 24, 192, 192, 576, 192, 1536, 576, 2688, 112, 896, 896, 2688, 416, 3328, 1728, 6784;
%e A160239 (End)
%e A160239 Right border gives A246030. - _Omar E. Pol_, Jan 29 2015 [This is simply a restatement of the theorem that this sequence is the Run Length Transform of A246030. - _N. J. A. Sloane_, Jan 29 2015]
%e A160239 .
%e A160239 From _Omar E. Pol_, Mar 18 2015 (Start):
%e A160239 Also, the sequence can be written as an irregular tetrahedron as shown below:
%e A160239 1;
%e A160239 ..
%e A160239 8;
%e A160239 ..
%e A160239 8;
%e A160239 24;
%e A160239 .........
%e A160239 8, 64;
%e A160239 24;
%e A160239 112;
%e A160239 ...................
%e A160239 8, 64, 64, 192;
%e A160239 24, 192;
%e A160239 112;
%e A160239 416;
%e A160239 .....................................
%e A160239 8, 64, 64, 192, 64, 512,192, 896;
%e A160239 24, 192, 192, 576;
%e A160239 112, 896;
%e A160239 416;
%e A160239 1728;
%e A160239 .......................................................................
%e A160239 8, 64, 64, 192, 64, 512,192, 896,64,512,512,1536,192,1536,896,3328;
%e A160239 24, 192, 192, 576,192,1536,576,2688;
%e A160239 112, 896, 896,2688;
%e A160239 416,3328;
%e A160239 1728;
%e A160239 6784;
%e A160239 ...
%e A160239 Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k). On the other hand, it appears that the configuration of ON cells of T(s,r,k) is also the central part of the configuration of ON cells of T(s+1,r+1,k).
%e A160239 (End)
%p A160239 # From _N. J. A. Sloane_, Jan 19 2015:
%p A160239 f:=proc(n) option remember;
%p A160239 if n=0 then RETURN(1);
%p A160239 elif n mod 2 = 0 then RETURN(f(n/2))
%p A160239 elif n mod 4 = 1 then RETURN(8*f((n-1)/4))
%p A160239 else RETURN(f(n-2)+2*f((n-1)/2)); fi;
%p A160239 end;
%p A160239 [seq(f(n),n=0..255)];
%t A160239 A160239[n_] :=
%t A160239 CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{n}}][[1]] // Total@*Total (* _Charles R Greathouse IV_, Aug 21 2014 *)
%t A160239 ArrayPlot /@ CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 30] (* _Charles R Greathouse IV_, Aug 21 2014 *)
%o A160239 (Haskell)
%o A160239 import Data.List (transpose)
%o A160239 a160239 n = a160239_list !! n
%o A160239 a160239_list = 1 : (concat $
%o A160239 transpose [a8, hs, zipWith (+) (map (* 2) hs) a8, tail a160239_list])
%o A160239 where a8 = map (* 8) a160239_list;
%o A160239 hs = h a160239_list; h (_:x:xs) = x : h xs
%o A160239 -- _Reinhard Zumkeller_, Feb 13 2015
%o A160239 (PARI) A160239=[];a(n)={if(n>#A160239,A160239=concat(A160239,vector(n-#A160239)),n||return(1);A160239[n]&&return(A160239[n]));A160239[n]=if(bittest(n,0),if(bittest(n,1),a(n-2)+2*a(n\2),a(n\4)*8),a(n\2))} \\ _M. F. Hasler_, May 10 2016
%Y A160239 Cf. A122108, A147562, A164032, A245180 (gives a(n)/8, n>=2).
%Y A160239 Cf. also A245542 (Partial sums), A245543, A083424, A245562, A246030, A254731 (an "even-rule" version).
%K A160239 nonn,nice
%O A160239 0,2
%A A160239 _John W. Layman_, May 05 2009
%E A160239 Offset changed to 1 by _Hrothgar_, Jul 11 2014
%E A160239 Offset reverted to 0 by _N. J. A. Sloane_, Jan 19 2015