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
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*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
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
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Dec 19 2004
EXTENSIONS
Edited by N. J. A. Sloane, Dec 05 2020
STATUS
approved