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

1,1

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

Table of n, a(n) for n=1..5.

Index entries for sequences related to cellular automata

Formula

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

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:

Crossrefs

A048706 gives the corresponding "XOR-conjugate" rules.

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

Sequence in context: A125010 A241004 A294310 * A119895 A270266 A255792

Adjacent sequences: A048702 A048703 A048704 * A048706 A048707 A048708

Keyword

nonn

Author

Antti Karttunen, Mar 09 1999

Status

approved