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%I #74 Jul 31 2022 07:49:29
%S 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,
%T 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,
%U 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
%N Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.
%C From _Franklin T. Adams-Watters_, Jul 05 2009: (Start)
%C Divided into rows of length 2n, row n consists of n 1's followed by n 0's.
%C Characteristic function of A061885, 1-based characteristic function of A004201. (End)
%C From _Wolfdieter Lang_, Dec 05 2012: (Start)
%C The row lengths sequence is A005408 (the odd numbers). The sum of row No. n is given by A000027(n+1).
%C 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.
%C 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.
%C (End)
%C This sequence is also generated by Rule 252. - _Robert Price_, Jan 31 2016
%C a(n) is 1 if the nearest square to n is >= n, otherwise 0. - _Branko Curgus_ Apr 25 1017
%H Robert Price, <a href="/A118175/b118175.txt">Table of n, a(n) for n = 0..9999</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule220.html">Rule 220</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F a(n) = 1 - A079813(n+1). - _Philippe Deléham_, Jan 02 2012
%F a(n) = 1 - ceiling(sqrt(n+1)) + round(sqrt(n+1)). - _Branko Curgus_, Apr 27 2017 [Corrected by _Ridouane Oudra_, Dec 01 2019]
%F 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
%F a(n) = floor(sqrt(n+1)+1/2) - floor(sqrt(n)) = round(sqrt(n+1)) - floor(sqrt(n)). - _Ridouane Oudra_, Dec 01 2019
%e The table a(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
%e 0: 1
%e 1: 0 1 1
%e 2: 0 0 1 1 1
%e 3: 0 0 0 1 1 1 1
%e 4: 0 0 0 0 1 1 1 1 1
%e 5: 0 0 0 0 0 1 1 1 1 1 1
%e 6: 0 0 0 0 0 0 1 1 1 1 1 1 1
%e 7: 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
%e 8: 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
%e 9: 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
%e ... Reformatted and extended by _Wolfdieter Lang_, Dec 2012
%e 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.
%e 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.
%e 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.
%t Table[1 - Ceiling[Sqrt[n]] + Round[Sqrt[n]], {n, 1, 257}] (* _Branko Curgus_, Apr 26 2017 *)
%t Table[{Array[1&,n],Array[0&,n]},{n,1,5}]//Flatten (* _Wolfgang Hintze_, Jul 28 2017 *)
%o (Python)
%o from math import isqrt
%o 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
%Y Cf. A083420, A219238.
%K nonn,tabf
%O 0,1
%A _Eric W. Weisstein_, Apr 13 2006
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