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 A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y). 13
 1, 9, 9, 25, 9, 81, 25, 121, 9, 81, 81, 225, 25, 225, 121, 441, 9, 81, 81, 225, 81, 729, 225, 1089, 25, 225, 225, 625, 121, 1089, 441, 1849, 9, 81, 81, 225, 81, 729, 225, 1089, 81, 729, 729, 2025, 225, 2025, 1089, 3969, 25, 225, 225, 625, 225, 2025, 625, 3025, 121, 1089, 1089, 3025, 441, 3969, 1849, 7225, 9, 81, 81, 225, 81, 729, 225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation. This is the odd-rule cellular automaton defined by OddRule 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). Run Length Transform of {A001045(k+2)^2} (or of A139818). 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..8192 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 [math.CO], 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 [math.CO], 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 [math.CO], 2015. FORMULA a(n) = A071053(n)^2. EXAMPLE Here is the neighborhood: [X, X, X] [X, X, X] [X, X, X] which contains a(1) = 9 ON cells. . From Omar E. Pol, Mar 17 2015: (Start) Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A139818(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below: .. 9; ... 9; 25; .......... 9,     81; 25; 121; .................... 9,     81,  81, 225; 25,   225; 121; 441; ........................................ 9,     81,  81, 225, 81, 729, 225, 1089; 25,   225, 225, 625; 121, 1089; 441; 1849; ... Note that every row r is equal to A139818(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number: T(s,r,k) = T(s+1,r,k). (End) MAPLE C:=f->subs({x=1, y=1}, f); # Find number of ON cells in CA for generations 0 thru M defined by rule # that cell is ON iff number of ON cells in nbd at time n-1 was odd # where nbd is defined by a polynomial or Laurent series f(x, y). OddCA:=proc(f, M) global C; local n, a, i, f2, p; f2:=simplify(expand(f)) mod 2; a:=[]; p:=1; for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od: lprint([seq(a[i], i=1..nops(a))]); end; f:=(1/x+1+x)*(1/y+1+y); OddCA(f, 70); MATHEMATICA b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)]; a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 30 2017 *) CROSSREFS Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034. Cf. A001045, A139818. Sequence in context: A144424 A282269 A205380 * A147340 A147499 A146591 Adjacent sequences:  A246032 A246033 A246034 * A246036 A246037 A246038 KEYWORD nonn AUTHOR N. J. A. Sloane, Aug 20 2014 STATUS approved

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Last modified October 21 13:24 EDT 2019. Contains 328299 sequences. (Running on oeis4.)