A057211 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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

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 *)

PROG

(Haskell)

a057211 n = a057211_list !! (n-1)

a057211_list = concat $ zipWith ($) (map replicate [1..]) a059841_list

-- Reinhard Zumkeller, Mar 18 2011

(Python)

from math import isqrt

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

CROSSREFS

Cf. A057212, A059841.

Sequence in context: A266982 A364746 A051341 * A120531 A106665 A004609

Adjacent sequences: A057208 A057209 A057210 * A057212 A057213 A057214

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

Definition amended by Georg Fischer, Oct 06 2021

STATUS

approved