A328106 Binary weight of A327971: a(n) = A000120(A110240(n) XOR A030101(A110240(n))).

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

Offset: 0

Author

Antti Karttunen, Oct 05 2019

Keywords

• nonn

Comments

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.

All terms are even.

Examples

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
---------------------------------------------- a(n)
   0:              (1)                          0
   1:             1(1)1                         0
   2:            11(0)01                        2
   3:           110(1)111                       2
   4:          1100(1)0001                      2
   5:         11011(1)10111                     4
   6:        110010(0)001001                    6
   7:       1101111(0)0111111                   4
   8:      11001000(1)11000001                  8
   9:     110111101(1)001000111                10
  10:    1100100001(0)1111011001               10
  11:   11011110011(0)10000101111               8
  12:  110010001110(0)110011010001             12
  13: 1101111011001(1)1011100110111             8
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16383
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to cellular automata

Crossrefs

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

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

Programs

PARI

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A328106(n) = hammingweight(bitxor(A110240(n), A030101(A110240(n))));
\\ Use this one for writing b-files:
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)); };

Formula

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

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