login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A102376 a(n) = 4^A000120(n). 41
1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 16, 64, 64, 256, 64, 256, 256, 1024, 64, 256, 256, 1024, 256, 1024, 1024 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Consider a simple cellular automaton, a grid of binary cells c(i,j), where the next state of the grid is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 2 If we start with a single cell having the value 1 and all the others 0, then the aggregate values of the subsequent states of the grid will be the terms in this sequence. - Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 31 2006. See link for initial states. - N. J. A. Sloane, Feb 12 2015

This is the odd-rule cellular automaton defined by OddRule 033 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015

First differences of A116520. - Omar E. Pol, May 05 2010

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

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.

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.

N. J. A. Sloane, Illustration of generations 0-15 of the cellular automaton

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

FORMULA

G.f.: product{k>=0, 1 + 4x^(2^k)}; a(n)=product{k=0..log_2(n), 4^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n; a(n)=sum{k=0..n, (C(n, k) mod 2)*3^A000120(n-k)}. (Formulas due to Paul D. Hanna.)

a(n) = sum{k=0..n, mod(C(n, k), 2)*sum{j=0..k, mod(C(k, j), 2)*sum{i=0..j, mod(C(j, i), 2)}}}. - Paul Barry, Apr 01 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u^2 - 2*u*v + 5*v^2) - 4*v^3. - Michael Somos, May 29 2008

Run length transform of A000302. - N. J. A. Sloane, Feb 23 2015

EXAMPLE

1 + 4*x + 4*x^2 + 16*x^3 + 4*x^4 + 16*x^5 + 16*x^6 + 64*x^7 + 4*x^8 + ...

From Omar E. Pol, Jun 07 2009: (Start)

Triangle begins:

1;

4;

4,16;

4,16,16,64;

4,16,16,64,16,64,64,256;

4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024;

4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024,16,64,64,256,64,256,...

(End)

MAPLE

seq(4^convert(convert(n, base, 2), `+`), n=0..100); # Robert Israel, Apr 30 2017

MATHEMATICA

Table[4^DigitCount[n, 2, 1], {n, 0, 100}] (* Indranil Ghosh, Apr 30 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, 4^subst( Pol( binary(n)), x, 1))} /* Michael Somos, May 29 2008 */

a(n) = 4^hammingweight(n); \\ Michel Marcus, Apr 30 2017

(Haskell)

a102376 = (4 ^) . a000120  -- Reinhard Zumkeller, Feb 13 2015

(Python)

def a(n): return 4**bin(n)[2:].count("1") # Indranil Ghosh, Apr 30 2017

CROSSREFS

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.

A151783 is a very similar sequence.

Cf. A001316, A048883, A000079, A116520, A000302.

See A160239 for the analogous CA defined by Rule 204 on an 8-celled neighborhood.

Sequence in context: A255300 A255298 A255302 * A217954 A273749 A273563

Adjacent sequences:  A102373 A102374 A102375 * A102377 A102378 A102379

KEYWORD

easy,nonn,tabf

AUTHOR

Paul Barry, Jan 05 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified January 20 02:38 EST 2018. Contains 297938 sequences.