0,2

Or, a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.

Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.

Fixed point of the morphism 1 -> 13, 3 -> 39, 9 -> 9(27), ... = 3^k -> 3^k 3^(k+1), ... starting from a(0) = 1; 1 -> 13 -> 1339 -> = 1339399(27) -> 1339399(27)399(27)9(27)(27)(81) -> ..., . - Robert G. Wilson v Jan 24 2006.

Equals row sums of triangle A166453, (the square of Sierpinski's gasket, A047999). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 13 2009]

T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176.

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Index entries for sequences related to cellular automata

O. E. Pol, Illustration of initial terms (Neighbors of the vertices) [From Omar E. Pol (info(AT)polprimos.com), Nov 06 2009]

O. E. Pol, Illustration of initial terms (Overlapping squares) [From Omar E. Pol (info(AT)polprimos.com), Nov 06 2009]

O. E. Pol, Illustration of initial terms (One-step bishop) [From Omar E. Pol (info(AT)polprimos.com), Nov 06 2009]

a(n)=product{k=0..log_2(n), 3^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n(offset 0). Formula from Paul D. Hanna.

a(n)=3a(n/2) if n is even, else a(n)=a((n+1)/2).

G.f.: Prod_{k>=0} (1+3*x^(2^k)). The generalization k^A000120 has generating function (1 + kx)(1 + kx^2)(1 + kx^4) ...

a(n+1)=sum(i=0, n, {binomial(n, i) (mod 2)}*sum(j=0, i, {binomial(i, j) (mod 2)})) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 16 2003

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)

Triangle begins:

..... 1;

..... 3;

..... 3,9;

..... 3,9,9,27;

..... 3,9,9,27,9,27,27,81;

..... 3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;

..... 3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27,...

Or

... 1;

... 3,3;

... 9,3,9,9;

.. 27,3,9,9,27,9,27,27;

.. 81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;

..243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27...

(End)

Nest[ Flatten[ # /. a_Integer -> {a, 3a}] &, {1}, 6] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006)

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

A generalization of A001316. Cf. A102376.

Partail sums give A130665. - David Applegate, Jun 11 2009

Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]

A166453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 13 2009]

Sequence in context: A165824 A151710 A160121 this_sequence A036553 A166466 A068219

Adjacent sequences: A048880 A048881 A048882 this_sequence A048884 A048885 A048886

nonn,nice,easy,new

John W. Layman (layman(AT)math.vt.edu)

Corrected by Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 19 2003

Entry revised by N. J. A. Sloane, May 30 2009. Offset changed to 0, Jun 11 2009