A327974 - OEIS

A327974

a(n) = A051023(n) XOR A051023(n-1), where A051023 gives the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0

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OFFSET

COMMENTS

Taking the first differences of indices of 1's in this sequence gives A327983 from its second term onward.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences related to cellular automata

Index entries for characteristic functions

FORMULA

a(n) = A051023(n) XOR A051023(n-1).

a(n) = A000035(floor(A327973(n) / A000079(n))).

EXAMPLE

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

0: (1) a(n)

1: 1(1)1 0

2: 11(0)01 1

3: 110(1)111 1

4: 1100(1)0001 0

5: 11011(1)10111 0

6: 110010(0)001001 1

7: 1101111(0)0111111 0

8: 11001000(1)11000001 1

9: 110111101(1)001000111 0

10: 1100100001(0)1111011001 1

11: 11011110011(0)10000101111 0

12: 110010001110(0)110011010001 0

13: 1101111011001(1)1011100110111 1

We start from row 1, and write 0 if the central cell is equal to the central cell in the row above, or 1 if it differs, which gives us terms: 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, ...

MATHEMATICA

A327974list[nmax_]:=BitXor@@@Partition[CellularAutomaton[30, {{1}, 0}, {nmax, {{0}}}], 2, 1]; A327974list[150] (* Paolo Xausa, May 26 2023 *)

PROG

(PARI)

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

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

A327973(n) = bitxor(A110240(n), 2*A110240(n-1));

A327974(n) = ((A327973(n)>>n)%2);

(PARI)

up_to = 105;

A269160(n) = bitxor(n, bitor(2*n, 4*n));

A327974list(up_to) = { my(v=vector(up_to), s=1, oc=s, nc, n=0, k=0); while(k<up_to, n++; s = A269160(s); nc = (s>>n)%2; k++; v[k] = bitxor(oc, nc); oc=nc); (v); }

v327974 = A327974list(up_to);

A327974(n) = v327974[n];

CROSSREFS

Cf. A000079, A003986, A003987, A110240, A269160, A327973, A327980, A327981, A327983.

Sequence in context: A188321 A257628 A203568 * A286049 A287657 A347198

Adjacent sequences: A327971 A327972 A327973 * A327975 A327976 A327977

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 03 2019

STATUS

approved