OFFSET
0,3
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (3,9,-27).
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)
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jul 27 2016
STATUS
approved