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 A255283 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-x-y. 2
 1, 7, 7, 31, 7, 49, 31, 145, 7, 49, 49, 217, 31, 217, 145, 601, 7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551, 7, 49, 49, 217, 49, 343, 217, 1015, 49, 343, 343, 1519, 217, 1519, 1015, 4207, 31, 217, 217, 961, 217, 1519, 961, 4495, 145, 1015, 1015, 4495, 601, 4207, 2551, 10351 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is the number of ON cells in a certain two-dimensional cellular automaton in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there were an odd number of ON cells in the neighborhood at the previous generation. This is the odd-rule cellular automaton defined by OddRule 537 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). LINKS 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 A255284. EXAMPLE Here is the neighborhood f: [X, 0, X] [X, X, 0] [X, X, X] which contains a(1) = 7 ON cells. From Omar E. Pol, Feb 22 2015: (Start) Written as an irregular triangle in which row lengths are the terms of A011782: 1; 7; 7, 31; 7, 49, 31, 145; 7, 49, 49, 217, 31, 217, 145, 601; 7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551; ... Right border gives: 1, 7, 31, 145, 601, 2551, ... This is simply a restatement of the theorem that this sequence is the Run Length Transform of A255284. (End) 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; .. 7; .. 7; 31; .......... 7,     49; 31; 145; ...................... 7,     49,   49,  217; 31,   217; 145; 601; ............................................ 7,     49,   49,  217,  49,  343, 217, 1015; 31,   217,  217,  961; 145, 1015; 601; 2551; ....................................................................................... 7,     49,   49,  217,  49,  343, 217, 1015, 49, 343, 343, 1519, 217, 1519, 1015, 4207; 31,   217,  217,  961, 217, 1519, 961, 4495; 145, 1015, 1015, 4495; 601, 4207; 2551; 10351; ... Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k). (End) MATHEMATICA (* f = A255284 *) f[n_] := If[EvenQ[n], 2^(2n+3)-5*7^(n/2), 2^(2n+3)-11*7^((n-1)/2)]/3; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *) CROSSREFS Cf. A255284. Sequence in context: A186142 A188274 A255281 * A140252 A095343 A286830 Adjacent sequences:  A255280 A255281 A255282 * A255284 A255285 A255286 KEYWORD nonn AUTHOR N. J. A. Sloane and Doron Zeilberger, Feb 19 2015 STATUS approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)