A267654 - OEIS

%I #19 Mar 04 2016 07:25:41

%S 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,

%T 12,10,12,10,12,10,12,14,12,14,12,14,12,14,12,14,12,14,12,14,16,14,16,

%U 14,16,14,16,14,16,14,16,14,16,14,16

%N 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.

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

%C Starting from A053186(n) =

%C 0,                            for b(n)

%C 0, 1, 2,                              for c(n)

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

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

%C etc,

%C a(n) is used for

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

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

%C 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).

%C Etc.

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

%C a(n) other writing (by pairs):

%C 2,   4,  2,  4,

%C 6,   4,  6,  4,

%C 6,   8,  6,  8,  6,  8,  6,  8,

%C 10   8, 10,  8, 10,  8, 10,  8,

%C 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12,

%C 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12,

%C etc.

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

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

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

%C Main diagonal: A005843(n+1).

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

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

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

%e The triangle is

%e 2,

%e 4, 2, 4,

%e 6, 4, 6, 4, 6,

%e 8, 6, 8, 6, 8, 6, 8,

%e etc.

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

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

%K nonn,tabf

%O 0,1

%A _Paul Curtz_, Jan 19 2016