A057211 Alternating runs of ones and zeros, where the n-th run has length n.

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

Offset: 1

Author

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

Keywords

• nonn
• tabl

Comments

Seen as a triangle read by rows: T(n,k) = n mod 2, 1<=k<=n. - Reinhard Zumkeller, Mar 18 2011

a(A007607(n)) = 0; a(A007606(n)) = 1. - Reinhard Zumkeller, Dec 30 2011

Row sums give A193356. - Omar E. Pol, Mar 05 2014

References

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

Links

• Reinhard Zumkeller, Rows n=1..125 of triangle, flattened
• Index entries for characteristic functions

Crossrefs

Cf. A057212, A059841.

Programs

Haskell

a057211 n = a057211_list !! (n-1)
a057211_list = concat $ zipWith ($) (map replicate [1..]) a059841_list
-- Reinhard Zumkeller, Mar 18 2011

Maple

A002024 := n->round(sqrt(2*n)):A057211 := n->(1-(-1)^A002024(n))/2;
# alternative Maple program:
T:= n-> [irem(n, 2)$n][]:
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 06 2021

Mathematica

Flatten[Table[{PadRight[{},n,1],PadRight[{},n+1,0]},{n,1,21,2}]] (* Harvey P. Dale, Jun 07 2015 *)

Python

from math import isqrt
def A057211(n): return int(bool(isqrt(n<<3)+1&2)) # Chai Wah Wu, Jun 19 2024

Formula

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

Also a(n) = A000035(A002024(n)) = A002024(n) mod 2 = A002024(n)-2*floor(A002024(n)/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003

G.f.: x/(1-x)*sum_{n>=0} (-1)^n*x^(n*(n+1)/2). - Mircea Merca, Mar 05 2014

a(n) = 1 - A057212(n). - Alois P. Heinz, Oct 06 2021

Extensions

Definition amended by Georg Fischer, Oct 06 2021