A071042 -id:A071042 - cp's OEIS frontend

A071038 Triangle read by rows giving successive states of cellular automaton generated by "Rule 182".

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

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.

			Triangle begins:
                           1,
                        1, 1, 1,
                     1, 0, 1, 0, 1,
                  1, 1, 1, 1, 1, 1, 1,
               1, 0, 1, 1, 1, 1, 1, 0, 1,
            1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1,
         1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
...

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Michael De Vlieger, Table of n, a(n) for n = 0..9999, i.e., rows 0 - 99.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

See A071042 for number of zeros at generation n.

Flatten@ Table[CellularAutomaton[182, {{1}, 0}, {{i}}], {i, 0, 9}] (* Michael De Vlieger, Mar 02 2015, based on N. J. A. Sloane at A071042 *)

Corrected by Hans Havermann, Jan 07 2012

A071055 Number of 0's in n-th row of triangle in A071038.

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 6, 0, 2, 2, 6, 2, 6, 6, 14, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6, 14, 14, 30, 6, 14, 14, 30, 14, 30, 30, 62, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6

Offset: 0

Hans Havermann, May 26 2002

nonn

a(n) is also the number of pairs of consecutive entries in the n-th row of Pascal's triangle with opposite parity.
All terms appear to be of the form 2^k - 2 (checked for n <= 10000). - Michael De Vlieger, Mar 02 2015
This appears to be equal to the number of previous values k, from 1..n-1, such that k AND n = k, where 'AND' is binary AND, and where the sequence starts at 1. For example, 1 AND 2 = 0, so a(2) = 0, while 1 AND 3 = 1 and 2 AND 3 = 2, so a(3) = 2. It follows from this that if n = 2^m - 1 then a(n) = n - 1 = 2^m - 2, giving the right border values noted below. - Scott R. Shannon, Apr 19 2023

			From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the powers of 2, the sequence begins:
0;
0,2;
0,2,2,6;
0,2,2,6,2,6,6,14;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30,2,6,6,14,6,14,14,30,6,14,14,30,14,30,30,62;
...
It appears that the right border gives the nonnegative terms of A000918.
It appears that the row sums give A056182.
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rule 182
Index entries for sequences related to cellular automata

Cf. A071042.

Count[#, n_ /; n == 0] & /@
Flatten[CellularAutomaton[182, {{1}, 0}, {{#}}] & /@ Range[0, 100],
1] (* Michael De Vlieger, Mar 02 2015 *)

A011371(n)=my(s); while(n>>=1, s+=n); s
a(n)=my(t=A011371(n)); sum(k=1,n,(A011371(k)+A011371(n-k)==t)!=(A011371(k-1)+A011371(n-k+1)==t)) \\ Charles R Greathouse IV, Mar 02 2015

a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+2-2[n==0] (conjectured). - Ralf Stephan, Mar 05 2004

a(n) = pext(n, n + 1) (conjectured) where pext is the "parallel bits extract" instruction of the x86 CPU; pext(x, mask) extracts bits from x at the bit locations specified by mask to contiguous low bits. - Falk Hüffner, Jul 26 2019

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A071038 Triangle read by rows giving successive states of cellular automaton generated by "Rule 182".

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