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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).

%I A327971 #29 Oct 06 2019 09:07:02
%S A327971 0,0,10,20,130,396,2842,4420,38610,124220,684490,1385044,8891330,
%T A327971 26281036,192525274,269101060,2454365330,8588410876,43860512138,
%U A327971 89059958420,551714970626,1663794165260,12235920695450,19683098342340,164315052318034,538162708968636,2894532467106378,6192136868790228,37503903254935874,114926395086966988,814341599153559130
%N 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).
%C A327971 Each term is a binary palindrome when its trailing zeros (in base 2) are omitted, that is, a term of A057890.
%C A327971 Compare the binary string illustrations drawn for the first 1024 terms of this sequence and for A327976, which has almost the same definition.
%H A327971 Antti Karttunen, <a href="/A327971/b327971.txt">Table of n, a(n) for n = 0..1023</a>
%H A327971 Antti Karttunen, <a href="/A327971/a327971.png">Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution</a>
%H A327971 Antti Karttunen, <a href="/A327971/a327971_1.png">Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution</a>
%H A327971 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A327971 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A327971 a(n) = A110240(n) XOR A265281(n).
%F A327971 a(n) = A280508(A110240(n)) = A110240(n) XOR A030101(A110240(n)).
%F A327971 a(n) = A280508(A265281(n)) = A265281(n) XOR A030101(A265281(n)).
%F A327971 For n >= 1, a(n) = (1/2) * (A327973(n-1) XOR A327976(n-1)).
%o A327971 (PARI)
%o A327971 A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
%o A327971 A110240(n) = if(!n,1,A269160(A110240(n-1)));
%o A327971 A269161(n) = bitxor(4*n, bitor(2*n, n));
%o A327971 A265281(n) = if(!n,1,A269161(A265281(n-1)));
%o A327971 A327971(n) = bitxor(A110240(n), A265281(n));
%o A327971 (PARI)
%o A327971 A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
%o A327971 A327971write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b327971.txt", n, " ", bitxor(s, A030101(s))); s = A269160(s)); };
%o A327971 (Python)
%o A327971 def A269160(n): return(n^((n<<1)|(n<<2)))
%o A327971 def A269161(n): return((n<<2)^((n<<1)|n))
%o A327971 def genA327971():
%o A327971     '''Yield successive terms of A327971.'''
%o A327971     s1 = 1
%o A327971     s2 = 1
%o A327971     while True:
%o A327971        yield (s1^s2)
%o A327971        s1 = A269160(s1)
%o A327971        s2 = A269161(s2)
%Y A327971 Cf. A003987, A030101, A057890, A110240, A265281, A280508, A328106 (binary weight of terms).
%Y A327971 Cf. also A327972, A327973, A327976, A328103, A328104 for other such combinations.
%K A327971 nonn
%O A327971 0,3
%A A327971 _Antti Karttunen_, Oct 03 2019