login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell. 4
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 0's for n > k and only 1's 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+1)) + round(sqrt(n+1)). - Branko Curgus, Apr 27 2017 [Corrected by Ridouane Oudra, Dec 01 2019]

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

a(n) = floor(sqrt(n+1)+1/2) - floor(sqrt(n)) = round(sqrt(n+1)) - floor(sqrt(n)). - Ridouane Oudra, Dec 01 2019

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 *)

PROG

(Python)

from math import isqrt

def A118175(n): return 1+int(n-(m:=isqrt(n+1))*(m+1)>=0)-int(m**2!=n+1) # Chai Wah Wu, Jul 30 2022

CROSSREFS

Cf. A083420, A219238.

Sequence in context: A267050 A267355 A070829 * A179762 A263804 A120526

Adjacent sequences: A118172 A118173 A118174 * A118176 A118177 A118178

KEYWORD

nonn,tabf

AUTHOR

Eric W. Weisstein, Apr 13 2006

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 09:30 EST 2022. Contains 358467 sequences. (Running on oeis4.)