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Binary weight of A327971: a(n) = A000120(A110240(n) XOR A030101(A110240(n))).
7

%I #20 Jun 09 2021 02:32:01

%S 0,0,2,2,2,4,6,4,8,10,10,8,12,8,18,6,12,26,16,18,14,18,20,22,22,26,26,

%T 38,30,26,36,26,28,36,28,18,28,42,36,32,34,40,44,38,40,50,48,48,50,58,

%U 46,56,48,42,54,48,56,56,46,54,48,52,60,58,78,74,64,60,66,74,74,64,80,74,80,62,92,62,80,70,68,100,90,82,80,92

%N Binary weight of A327971: a(n) = A000120(A110240(n) XOR A030101(A110240(n))).

%C a(n) is the number of times the k-th cell from the left is different from the k-th cell from the right, at the generation n of Rule 30 1-D cellular automaton, when it is started from a single alive cell.

%C All terms are even.

%H Antti Karttunen, <a href="/A328106/b328106.txt">Table of n, a(n) for n = 0..16383</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = A000120(A327971(n)) = A144078(A110240(n)) = A000120(A280508(A110240(n))).

%F a(n) = Sum_{i=0..2n} abs(A070950(n,i)-A070950(n,n-i)).

%e The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:

%e ---------------------------------------------- a(n)

%e 0: (1) 0

%e 1: 1(1)1 0

%e 2: 11(0)01 2

%e 3: 110(1)111 2

%e 4: 1100(1)0001 2

%e 5: 11011(1)10111 4

%e 6: 110010(0)001001 6

%e 7: 1101111(0)0111111 4

%e 8: 11001000(1)11000001 8

%e 9: 110111101(1)001000111 10

%e 10: 1100100001(0)1111011001 10

%e 11: 11011110011(0)10000101111 8

%e 12: 110010001110(0)110011010001 12

%e 13: 1101111011001(1)1011100110111 8

%e When we count the times the k-th cell from the left is different from the k-th cell from the right, we obtain a(n). Note that the central cells (indicated with parentheses) do not affect the count, as the central cell is always equal to itself.

%o (PARI)

%o A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.

%o A110240(n) = if(!n,1,A269160(A110240(n-1)));

%o A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));

%o A328106(n) = hammingweight(bitxor(A110240(n), A030101(A110240(n))));

%o \\ Use this one for writing b-files:

%o A328106write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b328106.txt", n, " ", hammingweight(bitxor(s, A030101(s)))); s = A269160(s)); };

%Y Cf. A000120, A070950, A110240, A144078, A269160, A280508, A327971.

%Y Cf. also A070952, A328105, A328107, A328108, A328109.

%K nonn

%O 0,3

%A _Antti Karttunen_, Oct 05 2019