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

0,1

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).

Links

Table of n, a(n) for n=0..63.

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, ....

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A214850 A236186 A143271 * A128859 A047975 A112791

Adjacent sequences: A267651 A267652 A267653 * A267655 A267656 A267657

Keyword

nonn,tabf

Author

Paul Curtz, Jan 19 2016

Status

approved