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

Author

N. J. A. Sloane, Jul 21 2014

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A038183, A189007.

Programs

C

#include 
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 */

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 *)

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).

Extensions

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