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

%I A048705 #14 Feb 08 2020 13:26:03
%S A048705 90,150,1721342310,140117185019831836588493434554119984790,
%T A048705 113427455640312821160607117168492587690
%N A048705 The rule numbers for 1-D CA composed of Rules "90" and "150" so that each direction occurs only once.
%C A048705 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.
%C A048705 The resulting numbers define one-dimensional linear cellular automata with radius being the sum of the number of the "90" and "150" components.
%C A048705 In hexadecimal the sequence is 5A, 96, 66999966, 69699696969669699696696969699696, 5555555555555555AAAAAAAAAAAAAAAA, ...
%H A048705 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A048705 a(n) = rule90x150combination(n) # See the Maple procedures below.
%p A048705 # The definitions of bit_i and floor_log_2 are given in A048700
%p A048705 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:
%p A048705 rule150 := proc(seed,n) option remember: local sl, i: if (0 = n) then (seed) else sl := floor_log_2(seed+1);
%p A048705 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:
%p A048705 # Rule 90 and Rule 150 are commutative in respect to each other:
%p A048705 rule90x150combination := proc(n) local p,q,i; p := extended_A020652[ n ]; # the Rule 150 component [ 0,1,op(A020652) ]
%p A048705 q := extended_A020653[ n ]; # the Rule 90 component [ 1,0,op(A020653) ]
%p A048705 RETURN(sum('bit_i(rule150(rule90(i,q),p),(2*(p+q))) * (2^i)','i'=0..(2^((2*(p+q))+1))-1));
%p A048705 end:
%Y A048705 A048706 gives the corresponding "XOR-conjugate" rules.
%Y A048705 Cf. A038183, A038184, A048709 (for specific examples). See also A048708, A048720.
%K A048705 nonn
%O A048705 1,1
%A A048705 _Antti Karttunen_, Mar 09 1999