A245191 - OEIS

%I #22 Jun 30 2018 21:16:33

%S 1,2,5,8,20,34,85,128,320,544,1360,2056,5140,8738,21845,32768,81920,

%T 139264,348160,526336,1315840,2236928,5592320,8388736,20971840,

%U 35652128,89130320,134744072,336860180,572662306,1431655765,2147483648,5368709120,9126805504

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

%C 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.

%C 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).

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

%C This is a one-sided version of A038183.

%H Lars Blomberg, <a href="/A245191/b245191.txt">Table of n, a(n) for n = 0..100</a>

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

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

%e Successive states are:

%e 1

%e 01

%e 101

%e 0001

%e 00101

%e 010001

%e 1010101

%e 00000001

%e 000000101

%e 0000010001

%e ...

%e which when converted from binary to decimal give the sequence.

%e This is the right-hand portion of the triangle in A038183.

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

%o (C)

%o #include <stdio.h>

%o int main()

%o {

%o         int u = 1, i = 1, n = 20;

%o         while (i++ <= n)

%o         {

%o                 printf("%d, ", u);

%o                 u = (u >> 1) ^ (u << 1);

%o         }

%o } /* _Luc Rousseau_, Jun 05 2018 */

%Y Cf. A038183, A189007.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jul 21 2014

%E Corrected a(11) and more terms from _Lars Blomberg_, Aug 08 2015