A327971 Bitwise XOR of trajectories of rule 30 and its mirror image, rule 86, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A265281(n).

0, 0, 10, 20, 130, 396, 2842, 4420, 38610, 124220, 684490, 1385044, 8891330, 26281036, 192525274, 269101060, 2454365330, 8588410876, 43860512138, 89059958420, 551714970626, 1663794165260, 12235920695450, 19683098342340, 164315052318034, 538162708968636, 2894532467106378, 6192136868790228, 37503903254935874, 114926395086966988, 814341599153559130

Offset

0,3

Comments

Each term is a binary palindrome when its trailing zeros (in base 2) are omitted, that is, a term of A057890.

Compare the binary string illustrations drawn for the first 1024 terms of this sequence and for A327976, which has almost the same definition.

Links

Antti Karttunen, Table of n, a(n) for n = 0..1023

Antti Karttunen, Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution

Antti Karttunen, Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution

Index entries for sequences related to binary expansion of n

Index entries for sequences related to cellular automata

Formula

a(n) = A110240(n) XOR A265281(n).

a(n) = A280508(A110240(n)) = A110240(n) XOR A030101(A110240(n)).

a(n) = A280508(A265281(n)) = A265281(n) XOR A030101(A265281(n)).

For n >= 1, a(n) = (1/2) * (A327973(n-1) XOR A327976(n-1)).

Prog

(PARI)
A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n, 1, A269160(A110240(n-1)));
A269161(n) = bitxor(4*n, bitor(2*n, n));
A265281(n) = if(!n, 1, A269161(A265281(n-1)));
A327971(n) = bitxor(A110240(n), A265281(n));
(PARI)
A030101(n) = if(n<1, 0, subst(Polrev(binary(n)), x, 2));
A327971write(up_to) = { my(s=1, n=0); for(n=0, up_to, write("b327971.txt", n, " ", bitxor(s, A030101(s))); s = A269160(s)); };
(Python)
def A269160(n): return(n^((n<<1)|(n<<2)))
def A269161(n): return((n<<2)^((n<<1)|n))
def genA327971():
    '''Yield successive terms of A327971.'''
    s1 = 1
    s2 = 1
    while True:
       yield (s1^s2)
       s1 = A269160(s1)
       s2 = A269161(s2)

Crossrefs

Cf. A003987, A030101, A057890, A110240, A265281, A280508, A328106 (binary weight of terms).

Cf. also A327972, A327973, A327976, A328103, A328104 for other such combinations.

Sequence in context: A220012 A154330 A006993 * A250107 A038693 A217318

Adjacent sequences: A327968 A327969 A327970 * A327972 A327973 A327974

Keyword

nonn

Author

Antti Karttunen, Oct 03 2019

Status

approved