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.
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
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
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, Apr 13 2006
STATUS
approved