This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A328106 #20 Jun 09 2021 02:32:01
%S A328106 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 A328106 38,30,26,36,26,28,36,28,18,28,42,36,32,34,40,44,38,40,50,48,48,50,58,
%U A328106 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 A328106 Binary weight of A327971: a(n) = A000120(A110240(n) XOR A030101(A110240(n))).
%C A328106 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 A328106 All terms are even.
%H A328106 Antti Karttunen, <a href="/A328106/b328106.txt">Table of n, a(n) for n = 0..16383</a>
%H A328106 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A328106 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A328106 a(n) = A000120(A327971(n)) = A144078(A110240(n)) = A000120(A280508(A110240(n))).
%F A328106 a(n) = Sum_{i=0..2n} abs(A070950(n,i)-A070950(n,n-i)).
%e A328106 The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
%e A328106 ---------------------------------------------- a(n)
%e A328106 0: (1) 0
%e A328106 1: 1(1)1 0
%e A328106 2: 11(0)01 2
%e A328106 3: 110(1)111 2
%e A328106 4: 1100(1)0001 2
%e A328106 5: 11011(1)10111 4
%e A328106 6: 110010(0)001001 6
%e A328106 7: 1101111(0)0111111 4
%e A328106 8: 11001000(1)11000001 8
%e A328106 9: 110111101(1)001000111 10
%e A328106 10: 1100100001(0)1111011001 10
%e A328106 11: 11011110011(0)10000101111 8
%e A328106 12: 110010001110(0)110011010001 12
%e A328106 13: 1101111011001(1)1011100110111 8
%e A328106 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 A328106 (PARI)
%o A328106 A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
%o A328106 A110240(n) = if(!n,1,A269160(A110240(n-1)));
%o A328106 A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
%o A328106 A328106(n) = hammingweight(bitxor(A110240(n), A030101(A110240(n))));
%o A328106 \\ Use this one for writing b-files:
%o A328106 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 A328106 Cf. A000120, A070950, A110240, A144078, A269160, A280508, A327971.
%Y A328106 Cf. also A070952, A328105, A328107, A328108, A328109.
%K A328106 nonn
%O A328106 0,3
%A A328106 _Antti Karttunen_, Oct 05 2019