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A101688
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Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0... and so on.
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15
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..101.
Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
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FORMULA
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G.f.: 1/[(1-xy)(1-y)]. k-th row of array: x^(k-1)/(1-x).
T(n, k) = if(binomial(k, n-k)>0, 1, 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+2) - 1/2). - Ridouane Oudra, Jul 16 2020
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Row sums of T (and antidiagonal sums of A) are A008619.
Cf. A079813, A168508.
Sequence in context: A087748 A117446 A187034 * A155029 A155031 A134540
Adjacent sequences: A101685 A101686 A101687 * A101689 A101690 A101691
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Dec 19 2004
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EXTENSIONS
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Edited by N. J. A. Sloane, Dec 05 2020
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STATUS
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approved
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