1,1

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, ...

Index entries for sequences related to cellular automata

a(n) = rule90x150combination(n) # See the Maple procedures below.

# 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:

A048706 gives the corresponding "XOR-conjugate" rules.

Cf. A038183, A038184, A048709 (for specific examples). See also A048708, A048720.

Sequence in context: A114826 A103653 A125010 this_sequence A119895 A119896 A044422

Adjacent sequences: A048702 A048703 A048704 this_sequence A048706 A048707 A048708

nonn

Antti Karttunen, Mar 09 1999