A274499 Sum of the degrees of asymmetry of all ternary words of length n.

0, 0, 6, 18, 108, 324, 1458, 4374, 17496, 52488, 196830, 590490, 2125764, 6377292, 22320522, 66961566, 229582512, 688747536, 2324522934, 6973568802, 23245229340, 69735688020, 230127770466, 690383311398, 2259436291848, 6778308875544, 22029503845518, 66088511536554

Offset: 0

Author

Emeric Deutsch, Jul 27 2016

Keywords

• nonn
• easy

Comments

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

A sequence is palindromic if and only if its degree of asymmetry is 0.

Examples

			a(2) = 6 because the ternary words 00, 01, 02, 10, 11, 12, 20, 21, 22 have degrees of asymmetry 0, 1, 1, 1, 0, 1, 1, 1, 0, respectively.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,9,-27).

Crossrefs

Cf. A274496, A274497, A274498.

Programs

Maple

a := proc (n) options operator, arrow: (1/6)*(2*n-1+(-1)^n)*3^n end proc: seq(a(n), n = 0 .. 30);

Mathematica

LinearRecurrence[{3, 9, -27}, {0, 0, 6}, 28] (* Jean-François Alcover, Sep 09 2024 *)

Formula

a(n) = (1/6)*(2n - 1 + (-1)^n)*3^n.

a(n) = Sum(k*A274498(n,k), k>=0).

From Chai Wah Wu, Dec 27 2018: (Start)

a(n) = 3*a(n-1) + 9*a(n-2) - 27*a(n-3) for n > 2.

G.f.: 6*x^2/((3*x - 1)^2*(3*x + 1)). (End)