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 A254460 a(n) is the number of predecessors of the all-ones state of the binary cellular automaton on the n X n grid graph with edges joining diagonal neighbors as well as vertical and horizontal neighbors, whose local rule is f(i,j) = sum of the state at vertex (i,j) and the states at all of its neighbors mod 2. 1
 1, 8, 1, 1, 512, 1, 1, 32768, 1, 1, 2097152, 1, 1, 134217728, 1, 1, 8589934592, 1, 1, 549755813888, 1, 1, 35184372088832, 1, 1, 2251799813685248, 1, 1, 144115188075855872, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence arose in a discussion among Carlos Rivera, Emmanuel Vantieghem, Dmitry Kamenetsky, W. Edwin Clark, Fred Schneider, RamÃ³n David, and Claudio Meller concerning Puzzle 772 at Prime Puzzles (see Prime Puzzle #772 link). Later we discovered the relationship to Sutner's paper. A corollary of that paper is that a(n) > 0 for all n. An obvious conjecture is that a(n) = 1 for n mod 3 = 0 or 1 and if n mod 3 = 2 then a(n) = 2^(2n-1). LINKS Carlos Rivera, Prime Puzzle #772 Klaus Sutner, Linear Cellular Automata and the Garden-of-Eden, Mathematical Intelligencer Vol 11, No. 2 1989. FORMULA Empirical g.f.: -x*(64*x^5+8*x^4+64*x^3-x^2-8*x-1) / ((x-1)*(4*x-1)*(x^2+x+1)*(16*x^2+4*x+1)). - Colin Barker, Jan 31 2015 EXAMPLE For n = 2 the a(2) = 8 predecessors of the all-ones matrix are the eight 2 X 2 binary matrices with one or three zero entries. MAPLE a:=proc(n) local A, A1, V, E, i, j, G, f, g, w; V:=NULL: E:={}: for i from 1 to n do for j from 1 to n do V:=V, [i, j]; E:=E union {seq(seq({[i, j], [i+x, j+y]}, x=-1..1), y=-1..1)}; od: od: V:=[V]; E:=remove(t->evalb(has(t, 0) or has(t, n+1)), E): E:=remove(t->evalb(nops(t) = 1), E): for i from 1 to nops(V)do    f(V[i]):=i: od: g:=proc(U)   map(f, U); end: G:=GraphTheory:-Graph(map(f, V), map(g, E)); A:=GraphTheory:-AdjacencyMatrix(G)+LinearAlgebra[IdentityMatrix](n^2); A1:=LinearAlgebra:-Modular:-Mod(2, convert(A, listlist), integer[]); w:=n^2-LinearAlgebra:-Modular:-Rank(2, A1); return 2^w; end proc: CROSSREFS Cf. A013713. Sequence in context: A178046 A176340 A111835 * A198830 A254244 A232068 Adjacent sequences:  A254457 A254458 A254459 * A254461 A254462 A254463 KEYWORD nonn AUTHOR W. Edwin Clark, Jan 30 2015 STATUS approved

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Last modified June 16 10:17 EDT 2021. Contains 345056 sequences. (Running on oeis4.)