A328106 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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.

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

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)).

EXAMPLE

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.

PROG

(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)); };

CROSSREFS

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

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

Sequence in context: A231515 A231544 A086420 * A342336 A320908 A356692