A071053 Number of ON cells at n-th generation of 1-D CA defined by Rule 150, starting with a single ON cell at generation 0.

1, 3, 3, 5, 3, 9, 5, 11, 3, 9, 9, 15, 5, 15, 11, 21, 3, 9, 9, 15, 9, 27, 15, 33, 5, 15, 15, 25, 11, 33, 21, 43, 3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, 45, 15, 45, 33, 63, 5, 15, 15, 25, 15, 45, 25, 55, 11, 33, 33, 55, 21, 63, 43, 85, 3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27

Offset

0,2

Comments

Number of 1's in n-th row of triangle in A071036.

Number of odd coefficients in (x^2+x+1)^n. - Benoit Cloitre, Sep 05 2003. This result was given in Wolfram (1983). - N. J. A. Sloane, Feb 17 2015

This is also the odd-rule cellular automaton defined by OddRule 007 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015

This is the Run Length Transform of S(n) = Jacobsthal(n+2) (cf. A001045). The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Sep 05 2014

References

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..16383 (first 1024 terms from T. D. Noe)

Joerg Arndt, A071053 (number of odd terms in expansion of (1+x+x^2)^n), SeqFan Mailing List, Mar 09 2015.

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Janko Gravner and Alexander E. Holroyd, Percolation and Disorder-Resistance in Cellular Automata, Annals of Probability, volume 43, number 4, 2015, pages 1731-1776.  Also arxiv:1304.7301 [math.PR], 2013-2015 and second author's copy. See Fig. 1.1 (left).

S. Kropf, S. Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016.

N. Pitsianis, P. Tsalides, G. L. Bleris, A. Thanailakis & H. C. Card, Deterministic one-dimensional cellular automata, Journal of Statistical Physics, 56(1-2), 99-112, 1989. [Discusses Rule 150]

T. Sillke and Helmut Postl, Odd trinomials: t(n) = (1+x+x^2)^n

T. Sillke and Helmut Postl, Odd trinomials: t(n) = (1+x+x^2)^n [Cached copy, with permission]

N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.

Index entries for sequences related to cellular automata

Formula

a(n) = Product_{i in row n of A245562} A001045(i+2) [Sillke]. For example, a(11) = A001045(3)*A001045(4) = 3*5 = 15. - N. J. A. Sloane, Aug 10 2014

Floor((a(n)-1)/4) mod 2 = A020987(n). - Ralf Stephan, Mar 18 2004

a(2*n) = a(n); a(2*n+1) = a(n) + 2*a(floor(n/2)). - Peter J. Taylor, Mar 26 2020

Example

May be arranged into blocks of sizes 1,1,2,4,8,16,...:
1,
3,
3, 5,
3, 9, 5, 11,
3, 9, 9, 15, 5, 15, 11, 21,
3, 9, 9, 15, 9, 27, 15, 33, 5, 15, 15, 25, 11, 33, 21, 43,
3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, 45, 15, 45, 33, 63, 5, 15, 15, 25, 15, 45, 25, 55, 11, 33, 33, 55, 21, 63, 43, 85,
3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, ...
... - N. J. A. Sloane, Sep 05 2014
.
From Omar E. Pol, Mar 15 2015: (Start)
Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A001045(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below (see also Joerg Arndt's equivalent program):
3;
..
3;
5;
.......
3,   9;
5;
11;
...............
3,   9,  9, 15;
5,  15;
11;
21;
...............................
3,   9,  9, 15,  9, 27, 15, 33;
5,  15, 15, 25;
11, 33;
21;
43;
..............................................................
3,   9,  9, 15,  9, 27, 15, 33, 9, 27, 27, 45, 15, 45, 33, 63;
5,  15, 15, 25, 15, 45, 25, 55;
11, 33, 33, 55;
21, 63;
43;
85;
...
Note that every row r is equal to A001045(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number.
(End)

Mathematica

a[n_] := Total[CoefficientList[(x^2 + x + 1)^n, x, Modulus -> 2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 05 2018 *)

Prog

(PARI)
b(n) = { (2^n - (-1)^n) / 3; }  \\ A001045
a(n)=
{
    if ( n==0, return(1) );
    \\ Use  a( 2^k * t ) = a(t)
    n \= 2^valuation(n, 2);
    if ( n==1, return(3) );  \\ Use a(2^k) == 3
    \\ now n is odd
    my ( v1 = valuation(n+1, 2) );
    \\ Use a( 2^k - 1 ) = A001045( 2 + k ):
    if ( n == 2^v1 - 1 ,  return( b( v1 + 2 ) ) );
    my( k2 = 1, k = 0 );
    while ( k2 < n,  k2 <<= 1; k+=1 );
    if ( k2 > n, k2 >>= 1; k-=1 );
    my( t = n - k2 );
    \\ here  n == 2^k + 1 where k maximal
    \\ Use the following:
    \\ a( 2^k + t ) =  3 * a(t)  if  t <= 2^(k-1)
    \\ a( 2^k + 2^(k-1) + t ) =  5 * a(t)  if  t <= 2^(k-2)
    \\ a( 2^k + 2^(k-1) + 2^(k-2) + t ) =  11* a(t)  if  t <= 2^(k-3)
    \\  ... etc. ...
    \\ a( 2^k + ... + 2^(k-s) + t ) = A001045(s+2) * a(t)  if  t <= 2^((k-1)-s)
    my ( s=1 );
    while ( 1 ,
        k2 >>= 1;
        if ( t <= k2 ,  return(  b(s+2) * a(t) ) );
        t -= k2;
        s += 1;
    );
}
\\ Joerg Arndt, Mar 15 2015, from SeqFan Mailing List, Mar 09 2015

Crossrefs

Cf. A038184, A118110, A071036, A001045, A253102, A020987, A246035, A245562.

Sequence in context: A048691 A332730 A248955 * A298398 A094439 A122037

Adjacent sequences: A071050 A071051 A071052 * A071054 A071055 A071056

Keyword

nonn,tabf

Author

Hans Havermann, May 26 2002

Extensions

Entry revised by N. J. A. Sloane, Aug 13 2014

Status

approved