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Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.
19

%I #81 Jun 25 2023 04:22:25

%S 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,

%T 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,

%U 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

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

%C 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:

%C 1

%C 0 1

%C 0 1 1

%C 0 0 1 1

%C 0 0 1 1 1

%C 0 0 0 1 1 1

%C ... The matrix T is used in A168508. [Comment revised by _N. J. A. Sloane_, Dec 05 2020]

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

%C For n >= 1, T(n,k) = number of partitions of n into k parts of sizes 1 or 2. - _Nicolae Boicu_, Aug 23 2018

%C 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

%D 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.

%H Boris Putievskiy, <a href="https://arxiv.org/abs/1212.2732">Transformations (of) Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.

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

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

%F T(n, k) = 1 if binomial(k, n-k) > 0, otherwise 0. - _Paul Barry_, Aug 23 2005

%F From _Boris Putievskiy_, Jan 09 2013: (Start)

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

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

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

%F a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n+1) - 1/2). - _Ridouane Oudra_, Jul 16 2020

%F a(n) = A103128(n+1) - A003056(n). - _Ridouane Oudra_, Apr 09 2022

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

%e The array A (on the left) and the triangle T of its antidiagonals (on the right):

%e 1 1 1 1 1 1 1 1 1 ......... 1

%e 0 1 1 1 1 1 1 1 1 ........ 0 1

%e 0 0 1 1 1 1 1 1 1 ....... 0 1 1

%e 0 0 0 1 1 1 1 1 1 ...... 0 0 1 1

%e 0 0 0 0 1 1 1 1 1 ..... 0 0 1 1 1

%e 0 0 0 0 0 1 1 1 1 .... 0 0 0 1 1 1

%e 0 0 0 0 0 0 1 1 1 ... 0 0 0 1 1 1 1

%e 0 0 0 0 0 0 0 1 1 .. 0 0 0 0 1 1 1 1

%e 0 0 0 0 0 0 0 0 1 . 0 0 0 0 1 1 1 1 1

%t 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 *)

%o (Python)

%o from math import isqrt

%o def A101688(n): return isqrt((m:=n<<1)+1)-(isqrt((m<<2)+8)+1>>1)+1 # _Chai Wah Wu_, Feb 10 2023

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

%Y Cf. A079813, A168508.

%Y Cf. A103128, A003056.

%K nonn,tabl

%O 0,1

%A _Ralf Stephan_, Dec 19 2004

%E Edited by _N. J. A. Sloane_, Dec 05 2020