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 A246031 Number of ON cells in 3-D cellular automaton described in Comments, after n generations. 4
 1, 26, 26, 124, 26, 676, 124, 1400, 26, 676, 676, 3224, 124, 3224, 1400, 10000, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 124, 3224, 3224, 15376, 1400, 36400, 10000, 89504, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 676, 17576, 17576, 83824, 3224, 83824, 36400, 260000, 124, 3224, 3224, 15376, 3224, 83824, 15376, 173600, 1400, 36400, 36400, 173600, 10000, 260000, 89504, 707008 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS We work on the cells of the 3-D grid. Each cell has 26 neighbors, A cell is ON iff an odd number of its neighbors were ON at the previous generation. We start with a single ON cell. 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1022 Shalosh B. Ekhad, Details about A246031 and A246032 Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package. Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015. 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: Part 1, Part 2 N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015 FORMULA This is the Run Length Transform of A246032 (see Comments). EXAMPLE The entries form blocks of sizes 1,1,2,4,8,...: 1, 26, 26, 124, 26, 676, 124, 1400, 26, 676, 676, 3224, 124, 3224, 1400, 10000, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 124, 3224, 3224, 15376, 1400, 36400, 10000, 89504, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 676, 17576, 17576, 83824, 3224, 83824, 36400, 260000, 124, 3224, 3224, 15376, 3224, 83824, 15376, 173600, 1400, 36400, 36400, 173600, 10000, 260000, 89504, 707008 ... From Omar E. Pol, Mar 19 2015: (Start) Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below: 1; .. 26; ... 26; 124; .......... 26,   676; 124; 1400; ..................... 26,   676, 676, 3224; 124,  3224; 1400; 10000; ............................................ 26,   676,  676, 3224, 676,17576,3224,36400; 124,  3224, 3224, 15376; 1400, 36400; 10000; 89504; .......................................................................................... 26,   676,  676, 3224, 676,17576,3224,36400,676,17576,17576,83824,3224,83824,36400,260000; 124,  3224, 3224, 15376, 3224, 83824, 15376, 173600; 1400, 36400, 36400, 173600; 10000, 260000; 89504; 707008; ... Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k). (End) MAPLE # This is a very inefficient program! f:=expand((1+x+x^2)*(1+y+y^2)*(1+z+z^2))-x*y*z; g:=n->expand(f^n) mod 2; h:=n->subs({x=1, y=1, z=1}, g(n)); [seq(h(n), n=0..30)]; # Better program from Roman Pearce, Feb 18 2015: f := Expand((1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z) mod 2: p := 1; for i from 1 to 100 do   p := Expand(p*f) mod 2;   lprint(nops(p)); end do: MATHEMATICA f = (1 + x + x^2)*(1 + y + y^2)*(1 + z + z^2) - x*y*z; p = 1; Print[1]; Join[{1}, Table[p = Expand[p*f] // PolynomialMod[#, 2]&; Lp = Length[p]; Print[Lp]; Lp, 100]] (* Jean-François Alcover, Jan 17 2018 *) PROG // MAGMA program from Roman Pearce, Feb 18 2015: P := PolynomialRing(GF(2), 3); f := (1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z; p := 1; for i := 1 to 100 do   p := p*f;   print(#Terms(p)); end for; CROSSREFS A 3-D analog of A160239 (2-D) and A255477 (4-D). Cf. A246032. Sequence in context: A003900 A040651 A022360 * A165847 A217981 A094835 Adjacent sequences:  A246028 A246029 A246030 * A246032 A246033 A246034 KEYWORD nonn AUTHOR N. J. A. Sloane, Aug 16 2014; corrected Aug 21 2014 STATUS approved

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Last modified May 26 10:08 EDT 2020. Contains 334620 sequences. (Running on oeis4.)