A245191 Successive states of one-sided one-dimensional cellular automaton using Rule 90, starting with a single ON cell, converted to decimal.

1, 2, 5, 8, 20, 34, 85, 128, 320, 544, 1360, 2056, 5140, 8738, 21845, 32768, 81920, 139264, 348160, 526336, 1315840, 2236928, 5592320, 8388736, 20971840, 35652128, 89130320, 134744072, 336860180, 572662306, 1431655765, 2147483648, 5368709120, 9126805504

Offset

0,2

Comments

The cells are labeled 0,1,2,3,4,5,6,... and we start at time 0 with cell 0 equal to 1, the rest 0.

At each step, the state of a cell is the mod 2 sum of the states of its left and right neighbors at the previous step (subject to the constraint that we only consider cells with nonnegative labels).

a(n) gives the state of the system, read from right to left, converted from binary to decimal.

This is a one-sided version of A038183.

Links

Lars Blomberg, Table of n, a(n) for n = 0..100

Formula

a(n) = Sum_{i>=0, i==n mod 2} (binomial(2n+2,n+2+i) mod 2)*2^i.

Write n = 2^k-1+j (k>=0, 0<=j<2^k). Then a(n) = 2^(k-j+1)*A038183(j).

Example

Successive states are:
1
01
101
0001
00101
010001
1010101
00000001
000000101
0000010001
...
which when converted from binary to decimal give the sequence.
This is the right-hand portion of the triangle in A038183.

Mathematica

a[ n_] := Sum[ Mod[Binomial[2 n + 2, n + i + 2], 2] 2^i, {i, Mod[n, 2], n}]; (* Michael Somos, Jun 30 2018 *)

Prog

(C)
#include <stdio.h>
int main()
{
        int u = 1, i = 1, n = 20;
        while (i++ <= n)
        {
                printf("%d, ", u);
                u = (u >> 1) ^ (u << 1);
        }
} /* Luc Rousseau, Jun 05 2018 */

Crossrefs

Cf. A038183, A189007.

Sequence in context: A054774 A360771 A104721 * A281104 A139407 A107384

Adjacent sequences: A245188 A245189 A245190 * A245192 A245193 A245194

Keyword

nonn

Author

N. J. A. Sloane, Jul 21 2014

Extensions

Corrected a(11) and more terms from Lars Blomberg, Aug 08 2015

Status

approved