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

Author

Ralf Stephan, Dec 19 2004

Keywords

• nonn
• tabl

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

Examples

			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
		

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

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

Crossrefs

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

Cf. A079813, A168508.

Cf. A103128, A003056.

Programs

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 *)

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

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

Extensions

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