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A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell. 3
1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

From Franklin T. Adams-Watters, Jul 05 2009: (Start)

Divided into rows of length 2n, row n consists of n 1's followed by n 0's.

Characteristic function of A061885, 1-based characteristic function of A004201. (End)

From Wolfdieter Lang, Dec 05 2012: (Start)

The row lengths sequence is A005408 (the odd numbers). The sum of row No. n is given by A000027(n+1).

This table is the first difference table of the q-binomial (Gauss polynomial) coefficient table G(2;n,k) = [q^k]( [n+2,2]_q) (see table A008967): a(n,k) = G(2;n,k) - G(2;n-1,k). The o.g.f. for the row polynomials is therefore G2(q,z) = (1-z)/product((1-q^j*z),j=0..2) = 1/((1-q*z)*(1-q^2*z)). Therefore, a(n,k) determines the number of partitions of k into precisely n parts, each <= 2. It determines also the number of partitions of k into at most 2 parts, each <= n but not <= (n-1), i.e., with part n present. See comments on A008967 regarding partitions.

From the o.g.f. G2(q,z) it should be clear that there are 0s for n > k and only 1s for k = n,...,2*n.

(End)

This sequence is also generated by Rule 252. - Robert Price, Jan 31 2016

a(n) is 1 if the nearest square to n is >= n, otherwise 0. - Branko Curgus Apr 25 1017

LINKS

Robert Price, Table of n, a(n) for n = 0..9999

Eric Weisstein's World of Mathematics, Rule 220

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

FORMULA

a(n) = 1 - A079813(n+1). - Philippe Deléham, Jan 02 2012

a(n) = 1 - ceiling(sqrt(n)) + round(sqrt(n)). - Branko Curgus, April 27 2017

G.f.: x/(1 - x)*( Sum_{n >= 1} x^(n^2-n)*(1-x^n)) =  1/(2-2x)* ( x + x^(3/4)*EllipticTheta(2,0,x) - x*EllipticTheta(3,0,x) ). - Wolfgang Hintze, Jul 28 2017

EXAMPLE

The table a(n,k) begins:

n\k 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...

0:  1

1:  0  1  1

2:  0  0  1  1  1

3:  0  0  0  1  1  1  1

4:  0  0  0  0  1  1  1  1  1

5:  0  0  0  0  0  1  1  1  1  1  1

6:  0  0  0  0  0  0  1  1  1  1  1  1  1

7:  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1

8:  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1

9:  0  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1  1

... Reformatted and extended by Wolfdieter Lang, Dec 2012

Partition examples: a(n,k) = 0 if n>k because the maximal number of parts of a partition of k is k. a(n,n) = 1, n >= 1, because only the partition 1^n has n parts, and 1 <= 2.

  a(2,3) = 1 because the only partition of 3 with 2 parts, each <= 2, is 1,2. Also, the only partition of 3 with at most 2 parts, each <= 2, and a part 2 present is also 1,2.

  a(5,7) =1 because the only 5-part partition of 7 with maximal part 2 is 1^3,2^3. Also, the only partition of 7 with at most 2 parts, each <= 5, which a part 5 present is 2,5.

MATHEMATICA

Table[1 - Ceiling[Sqrt[n]] + Round[Sqrt[n]], {n, 1, 257}] (* Branko Curgus, Apr 26 2017 *)

Table[{Array[1&, n], Array[0&, n]}, {n, 1, 5}]//Flatten (* Wolfgang Hintze, Jul 28 2017 *)

CROSSREFS

Cf. A083420, A219238.

Sequence in context: A108882 A168002 A070829 * A179762 A120526 A086694

Adjacent sequences:  A118172 A118173 A118174 * A118176 A118177 A118178

KEYWORD

nonn,tabf

AUTHOR

Eric W. Weisstein, Apr 13 2006

STATUS

approved

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Last modified September 21 15:13 EDT 2017. Contains 292300 sequences.