A267654 -id:A267654 - cp's OEIS frontend

A272188 Triangle with 2*n+1 terms per row, read by rows: the first row is 1 (by decree), following rows contain 0 to 2n+1 but omitting 2n.

1, 0, 1, 3, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 7, 0, 1, 2, 3, 4, 5, 6, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17

Offset: 0

Paul Curtz, Apr 22 2016

Row n is row 2n+1 of A128138, a bisection.
The second bisection by rows
0, 2,
0, 1, 2, 4,
0, 1, 2, 3, 4, 6,
0, 1, 2, 3, 4, 5, 6, 8,
etc
is the basis of
0, 2, 4, 6, 8, 10, 12, ... the even numbers A005843(n)
0, 1, 2, 4, 3,  6,  8, 5, 10, ... a permutation of the nonnegative integers A265667(n).
0, 1, 2, 3, 4,  6,  5, 8,  7, 10, 12, ... a permutation of the nonnegative integers A265734(n)
etc.
A005843(n) - A005843(n-1) = 2, for n>0.
A265667(n) - A265667(n-3) = 4, 2, 4 (period 3), for n>2.
A265734(n) - A265734(n-5) = 6, 4, 6, 4, 6 (period 5), for n>4.
See A267654.
For
1, 3, 5, 7, 9, 11, 13 ... the odd numbers A005408(n),
0, 1, 3, 2, 5,  7,  4, 9, 11, ... a permutation of the nonnegative numbers A006369,
0, 1, 2, 3, 5,  4,  7, 6,  9, 11, 8, 13, 10, 15, ... another permutation,
a(n) must be extended with one term by row:
1, 3,
0, 1, 3, 2,
0, 1, 2, 3, 5, 4,

			Irregular triangle:
1,
0, 1, 3,
0, 1, 2, 3, 5,
0, 1, 2, 3, 4, 5, 7,
0, 1, 2, 3, 4, 5, 6, 7, 9,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11,
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13,
etc.

Cf. A001477, A005408, A005843, A006369, A028310 (main diagonal), A053186, A265667, A265734, A267654.

Table[Delete[Range[0, 2 n + 1], 2 n + 1], {n, 0, 8}] // Flatten (* Michael De Vlieger, Apr 25 2016 *)

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A272188 Triangle with 2*n+1 terms per row, read by rows: the first row is 1 (by decree), following rows contain 0 to 2n+1 but omitting 2n.

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