A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.

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

Offset

0,1

Comments

The definition is that of a linear sequence. Equivalently, define a (0,1) infinite lower triangular matrix T(n,k) (0 <= k <= n) by T(n,k) = 1 if k >= n/2, 0 otherwise, and read it by rows. The triangle T begins:

1

0 1

0 1 1

0 0 1 1

0 0 1 1 1

0 0 0 1 1 1

... The matrix T is used in A168508. [Comment revised by N. J. A. Sloane, Dec 05 2020]

Also, square array A read by antidiagonals upwards: A(n,k) = 1 if k >= n, 0 otherwise.

For n >= 1, T(n,k) = number of partitions of n into k parts of sizes 1 or 2. - Nicolae Boicu, Aug 23 2018

T(n, k) is the number of ways to distribute n balls to k unlabeled urns in such a way that no urn receives more than one ball (see Beeler). - Stefano Spezia, Jun 16 2023

References

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Proposition 4.2.1 at p. 98.

Links

Table of n, a(n) for n=0..101.

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

G.f.: 1/((1 - x*y)*(1 - y)).

G.f. of k-th row of the array: x^(k-1)/(1 - x).

T(n, k) = 1 if binomial(k, n-k) > 0, otherwise 0. - Paul Barry, Aug 23 2005

From Boris Putievskiy, Jan 09 2013: (Start)

a(n) = floor((2*A002260(n)+1)/A003056(n)+3).

a(n) = floor((2*n-t*(t+1)+1)/(t+3)), where

t = floor((-1+sqrt(8*n-7))/2). (End)

a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n+1) - 1/2). - Ridouane Oudra, Jul 16 2020

a(n) = A103128(n+1) - A003056(n). - Ridouane Oudra, Apr 09 2022

E.g.f. of k-th column of the array: exp(x)*Gamma(1+k, x)/k!. - Stefano Spezia, Jun 16 2023

Example

The array A (on the left) and the triangle T of its antidiagonals (on the right):
  1 1 1 1 1 1 1 1 1 ......... 1
  0 1 1 1 1 1 1 1 1 ........ 0 1
  0 0 1 1 1 1 1 1 1 ....... 0 1 1
  0 0 0 1 1 1 1 1 1 ...... 0 0 1 1
  0 0 0 0 1 1 1 1 1 ..... 0 0 1 1 1
  0 0 0 0 0 1 1 1 1 .... 0 0 0 1 1 1
  0 0 0 0 0 0 1 1 1 ... 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 1 1 .. 0 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 0 1 . 0 0 0 0 1 1 1 1 1

Mathematica

rows = 15; A = Array[If[#1 <= #2, 1, 0]&, {rows, rows}]; Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, May 04 2017 *)

Prog

(Python)
from math import isqrt
def A101688(n): return isqrt((m:=n<<1)+1)-(isqrt((m<<2)+8)+1>>1)+1 # Chai Wah Wu, Feb 10 2023

Crossrefs

Row sums of T (and antidiagonal sums of A) are A008619.

Cf. A079813, A168508.

Cf. A103128, A003056.

Sequence in context: A087748 A117446 A187034 * A155029 A155031 A134540

Adjacent sequences: A101685 A101686 A101687 * A101689 A101690 A101691

Keyword

nonn,tabl

Author

Ralf Stephan, Dec 19 2004

Extensions

Edited by N. J. A. Sloane, Dec 05 2020

Status

approved