A048705 The rule numbers for 1-D CA composed of Rules "90" and "150" so that each direction occurs only once.

90, 150, 1721342310, 140117185019831836588493434554119984790, 113427455640312821160607117168492587690

Offset: 1

Author

Antti Karttunen, Mar 09 1999

Keywords

• nonn

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 number of the "90" and "150" components.

In hexadecimal the sequence is 5A, 96, 66999966, 69699696969669699696696969699696, 5555555555555555AAAAAAAAAAAAAAAA, ...

Links

• Index entries for sequences related to cellular automata

Crossrefs

A048706 gives the corresponding "XOR-conjugate" rules.

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

Programs

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:

Formula

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