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

0,4

Comments

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,

Links

Table of n, a(n) for n=0..80.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A235919 A229654 A306288 * A049765 A343394 A194801

Adjacent sequences: A272185 A272186 A272187 * A272189 A272190 A272191

Keyword

nonn,tabf,easy

Author

Paul Curtz, Apr 22 2016

Status

approved