cp's OEIS Frontend

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

%I A101688 #81 Jun 25 2023 04:22:25
%S A101688 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 A101688 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 A101688 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 A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.
%C A101688 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 A101688   1
%C A101688   0 1
%C A101688   0 1 1
%C A101688   0 0 1 1
%C A101688   0 0 1 1 1
%C A101688   0 0 0 1 1 1
%C A101688 ...  The matrix T is used in A168508. [Comment revised by _N. J. A. Sloane_, Dec 05 2020]
%C A101688 Also, square array A read by antidiagonals upwards: A(n,k) = 1 if k >= n, 0 otherwise.
%C A101688 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 A101688 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 A101688 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 A101688 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 A101688 G.f.: 1/((1 - x*y)*(1 - y)).
%F A101688 G.f. of k-th row of the array: x^(k-1)/(1 - x).
%F A101688 T(n, k) = 1 if binomial(k, n-k) > 0, otherwise 0. - _Paul Barry_, Aug 23 2005
%F A101688 From _Boris Putievskiy_, Jan 09 2013: (Start)
%F A101688 a(n) = floor((2*A002260(n)+1)/A003056(n)+3).
%F A101688 a(n) = floor((2*n-t*(t+1)+1)/(t+3)), where
%F A101688 t = floor((-1+sqrt(8*n-7))/2). (End)
%F A101688 a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n+1) - 1/2). - _Ridouane Oudra_, Jul 16 2020
%F A101688 a(n) = A103128(n+1) - A003056(n). - _Ridouane Oudra_, Apr 09 2022
%F A101688 E.g.f. of k-th column of the array: exp(x)*Gamma(1+k, x)/k!. - _Stefano Spezia_, Jun 16 2023
%e A101688 The array A (on the left) and the triangle T of its antidiagonals (on the right):
%e A101688   1 1 1 1 1 1 1 1 1 ......... 1
%e A101688   0 1 1 1 1 1 1 1 1 ........ 0 1
%e A101688   0 0 1 1 1 1 1 1 1 ....... 0 1 1
%e A101688   0 0 0 1 1 1 1 1 1 ...... 0 0 1 1
%e A101688   0 0 0 0 1 1 1 1 1 ..... 0 0 1 1 1
%e A101688   0 0 0 0 0 1 1 1 1 .... 0 0 0 1 1 1
%e A101688   0 0 0 0 0 0 1 1 1 ... 0 0 0 1 1 1 1
%e A101688   0 0 0 0 0 0 0 1 1 .. 0 0 0 0 1 1 1 1
%e A101688   0 0 0 0 0 0 0 0 1 . 0 0 0 0 1 1 1 1 1
%t A101688 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 A101688 (Python)
%o A101688 from math import isqrt
%o A101688 def A101688(n): return isqrt((m:=n<<1)+1)-(isqrt((m<<2)+8)+1>>1)+1 # _Chai Wah Wu_, Feb 10 2023
%Y A101688 Row sums of T (and antidiagonal sums of A) are A008619.
%Y A101688 Cf. A079813, A168508.
%Y A101688 Cf. A103128, A003056.
%K A101688 nonn,tabl
%O A101688 0,1
%A A101688 _Ralf Stephan_, Dec 19 2004
%E A101688 Edited by _N. J. A. Sloane_, Dec 05 2020