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A138277
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Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4 (with a single 1 as initial condition).
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2
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1, 5, 13, 49, 109, 473, 1081, 4037, 8749, 37913, 88465, 325021, 717337, 3108461, 7095613, 26490289, 57395629, 248714393, 580333585, 2132141341, 4707150193, 20397650837, 46548642709, 173816036825, 376630110937, 1632063814061, 3808148899477, 13991111158153
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OFFSET
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0,2
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COMMENTS
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See A138276 for the corresponding sequence for a Bethe lattice with coordination number 3.
See A001045 for the corresponding sequence on a 1D lattice (equivalent to a k=2 Bethe lattice); this is based on the Jacobsthal sequence A001045.
See A072272 for the corresponding sequence on a 2D lattice (based on A007483).
Related to Cellular Automata.
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LINKS
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FORMULA
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The total number of nodes in state 1 after n iterations (starting with a single 1) of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4. Rule 150 sums the values of the focal node and its k neighbors, then applies modulo 2.
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EXAMPLE
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Let x_0 be the state (0 or 1) of the focal node and x_i the state of every node that is i steps away from the focal node. In time step n=0, all x_i=0 except x_0=1 (start with a single seed). In the next step, x_1=1 as they have 1 neighbor being 1. For n=2, the x_1 nodes have 1 neighbor being 1 (x_0) and themselves being 1; the sum being 2, modulo 2, resulting in x_1=0.
The focal node and outmost nodes x_n are always 1.
Thus one has the patterns
x_0, x_1, x_2, ...
1
1 1
1 0 1
1 0 1 1
1 0 0 0 1
1 1 0 1 1 1
1 0 0 0 1 0 1
1 1 0 1 1 0 1 1
1 0 0 0 0 0 0 0 1
(N.B.: This is equivalent to the right half plane of Rule 150 in 1D.)
The nodes have the multiplicities 1,4,12,36,108,324,972,...
The sequence then is obtained by
a(n)= x_0(n) + 4*(x_1(n) + sum_(i=2...n) x_i(n) * 3^(i-1)).
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MATHEMATICA
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nmax = 30;
states = CellularAutomaton[150, {{1}, 0}, nmax];
T[n_, i_] := states[[n+1, nmax+i+1]];
a[n_] := T[n, 0] + 4(T[n, 1]+Sum[3^(i-1) T[n, i], {i, 2, n}]);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Jens Christian Claussen (claussen(AT)theo-physik.uni-kiel.de), Mar 11 2008
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EXTENSIONS
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STATUS
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approved
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