

A048883


a(n) = 3^wt(n), where wt(n) = A000120(n).


53



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, 81, 81, 243, 81, 243, 243, 729, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81
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OFFSET

0,2


COMMENTS

Or, a(n)=number of 1's ("live" cells) at stage n of a 2dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.
This is the oddrule cellular automaton defined by OddRule 013 (see EkhadSloaneZeilberger "OddRule Cellular Automata on the Square Grid" link).  N. J. A. Sloane, Feb 25 2015
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
First bisection of A169697=1,5,3,19,3,. a(2n+2)+a(2n+3)=12,12,36,=12*A147610 ? Distribution of terms (in A000244): A011782=1,A000079 for first array, A000079 for second.  Paul Curtz, Apr 20 2010
This sequence pertains to phenotype Punnett square mathematics. Start with X=1. Each hybrid cross involves the equation X:3X. Therefore, the ratio in the first (mono) hybrid cross is X=1:3X=3(1) or 3; or 3:1. When you move up to the next hybridization level, replace the previous cross ratio with X. X now represents 2 numbers1:3. Therefore, the ratio in the second (di) hybrid cross is X=(1:3):3X=[3(1):3(3)] or (3:9). Put it together and you get 1:3:3:9. Each time you move up a hybridization level, replace the previous ratio with X, and use the same equationX:3X to get its ratio.  John Michael Feuk, Dec 10 2011


LINKS

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


FORMULA

a(n) = Product_{k=0..log_2(n)} 3^b(n,k), where b(n,k) = coefficient of 2^k in binary expansion of n (offset 0).  Paul D. Hanna
a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
G.f.: Product_{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, Nov 16 2003
G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2).  Ilya Gutkovskiy, Jul 09 2019


EXAMPLE

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)


MATHEMATICA

Nest[ Join[#, 3#] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014*)


PROG

(PARI) a(n)=n=binary(n); 3^sum(i=1, #n, n[i])
(Haskell)


CROSSREFS

For generating functions Product_{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.


KEYWORD



AUTHOR



EXTENSIONS

Offset changed to 0, Jun 11 2009


STATUS

approved



