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COMMENTS
| The "numerator" (0, 1 and the rest from A020652) is the multiplicity of the "Rule 150" component and the "denominator" (1, 0 and the rest from A020653) is the multiplicity of the "Rule 90" component.
The resulting numbers define one-dimensional linear cellular automata with radius being the sum of the amount of the "90" and "150" components.
In hexadecimal the sequence is 5A, 96, 66999966, 69699696969669699696696969699696, 5555555555555555AAAAAAAAAAAAAAAA, ...
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MAPLE
| # The definitions of bit_i and floor_log_2 are given in A048700
rule90 := proc(seed, n) option remember: local sl, i: if (0 = n) then (seed) else sl := floor_log_2(seed+1); add(((bit_i(rule90(seed, n-1), i)+bit_i(rule90(seed, n-1), i-2)) mod 2)*(2^i), i=0..(2*n)+sl) fi: end:
rule150 := proc(seed, n) option remember: local sl, i: if (0 = n) then (seed) else sl := floor_log_2(seed+1);
add(((bit_i(rule150(seed, n-1), i)+bit_i(rule150(seed, n-1), i-1)+bit_i(rule150(seed, n-1), i-2)) mod 2)*(2^i), i=0..((2*n)+sl)) fi: end:
# Rule 90 and Rule 150 are commutative in respect to each other:
rule90x150combination := proc(n) local p, q, i; p := extended_A020652[ n ]; # the Rule 150 component [ 0, 1, op(A020652) ]
q := extended_A020653[ n ]; # the Rule 90 component [ 1, 0, op(A020653) ]
RETURN(sum('bit_i(rule150(rule90(i, q), p), (2*(p+q))) * (2^i)', 'i'=0..(2^((2*(p+q))+1))-1));
end:
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