A057212 n-th run has length n.

0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 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, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Ben Tyner (tyner(AT)phys.ufl.edu), Sep 27 2000

Keywords

• easy
• nonn
• tabl

Comments

T(n,k) = 1 - n mod 2, 1 <= k <= n. [Reinhard Zumkeller, Mar 18 2011]

{a(n)} interpreted as a string over {0,1} is one of exactly two fixed-points of the function defined by f(0^n 1 s) = 1^(n-1) f(1 s) and f(1^n 0 s) = 0^(n-1) f(0 s). The other fixed point is obtained by swapping all 0s and 1s. - Curtis Bechtel, Jun 27 2025

References

• K. H. Rosen, Discrete Mathematics and its Applications, 1999, fourth edition, page 79, exercise 10 (g).

Crossrefs

Cf. A057211.

As a simple triangular or square array virtually the only sequences which appear are A000004, A000012 and A000035. Cf. A060510.

Programs

Haskell

a057212 n = a057212_list !! (n-1)
a057212_list = concat $ zipWith ($) (map replicate [1..]) a000035_list
-- Reinhard Zumkeller, Mar 18 2011

Maple

A002024 := n->round(sqrt(2*n)):A057212 := n->(1+(-1)^A002024(n))/2;
# alternative Maple program:
T:= n-> [irem(1+n, 2)$n][]:
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 06 2021

Mathematica

Table[If[OddQ[n], 0, 1], {n, 1, 14}, {n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

Python

from math import isqrt
def A057212(n): return int(not isqrt(n<<3)+1&2) # Chai Wah Wu, Jun 19 2024

Formula

a(n)=A003056(n) mod 2 so as a square array T(n, k)=n+k mod 2 - Henry Bottomley, Mar 22 2001

a(n) = (1+(-1)^A002024(n))/2, where A002024(n)=round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003

a(n)=A163334(n) mod 2 = A163336(n) mod 2 = A163357(n) mod 2 = A163359(n) mod 2, i.e. the array gives the parity of elements at the successive antidiagonals (alternating between 0 and 1) of square arrays constructed from ANY Hilbert curve starting from zero located at the top left corner of a square grid (and using only N,E,S,W steps of length one). - Antti Karttunen, Oct 22 2012

a(n) = 1 - A057211(n). - Alois P. Heinz, Oct 06 2021