A267654 Irregular triangle of palindromic subsequences. Every row has 2*n+1 terms. From the second row, there are only two alternated numbers: 2*n+4 and 2*n+2.

2, 4, 2, 4, 6, 4, 6, 4, 6, 8, 6, 8, 6, 8, 6, 8, 10, 8, 10, 8, 10, 8, 10, 8, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16

Offset: 0

Author

Paul Curtz, Jan 19 2016

Keywords

• nonn
• tabf

Comments

Row sums = 2, 10, 26, 50, ... = A069894(n).

Starting from A053186(n) =

0, for b(n)

0, 1, 2, for c(n)

0, 1, 2, 3, 4, for d(n)

0, 1, 2, 3, 4, 5, 6,

etc,

a(n) is used for

1) b(n+1) = b(n) + (a(0)=2) i.e. 0, 2, 4, 6, ... = A005843(n).

2) c(n+3) = c(n) + (period 3:repeat 4, 2, 4) i.e. 0, 1, 2, 4, 3, 6, 8, ... = A265667(n).

3) d(n+5) = d(n) + (period 5:repeat 6, 4, 6, 4, 6) i.e. 0, 1, 2, 3, 4, 6, 5, 8, 7, 10, ... = A265734(n).

Etc.

a(n) has a companion with the same terms,differently distributed,yielding permutations of the nonnegative numbers. See A265672.

a(n) other writing (by pairs):

2, 4, 2, 4,

6, 4, 6, 4,

6, 8, 6, 8, 6, 8, 6, 8,

10 8, 10, 8, 10, 8, 10, 8,

10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12,

14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12,

etc.

First column: A168276(n+2). Second column: A168273(n+2).

Row sums: 12, 20, 56, 72, ... = 4*A074378(n+1).

The last term of the successive rows is the number of their terms.

Main diagonal: A005843(n+1).

Examples

			The triangle is
2,
4, 2, 4,
6, 4, 6, 4, 6,
8, 6, 8, 6, 8, 6, 8,
etc.
		

Crossrefs

Cf. A007395, A005843, A008586, A016825, A053186, A069894, A074378, A086520, A168273, A168276, A265667, A265672, A265734.

Programs

Mathematica

Table[2 (n - 1) + 2 (Boole@ OddQ@ k + 1), {n, 0, 7}, {k, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 19 2016 *)

Formula

a(n) = 2 * A086520(n+2).

a(2n) = 4*n + 2 times 4*n + 2 = 2, 2, 6, 6, 6, 6, 6, 6, 10,....

a(2n+1) = 4*(n+1) times 4*(n+1) = 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 12, ....