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
FORMULA
a(n) = (1/8)*(2n - 1 + (-1)^n)*2^n.
a(n) = Sum_{k>=0} k*A274496(n,k).
From Alois P. Heinz, Jul 27 2016: (Start)
a(n) = 2^(n-1) * A004526(n) = 2^(n-1)*floor(n/2).
a(n) = 2 * A134353(n-2) for n>=2. (End)
From Chai Wah Wu, Dec 27 2018: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) for n > 2.
G.f.: 2*x^2/((2*x - 1)^2*(2*x + 1)). (End)
EXAMPLE
a(3) = 4 because the binary words 000, 001, 010, 100, 011, 101, 110, 111 have degrees of asymmetry 0, 1, 0, 1, 1, 0, 1, 0, respectively.
MAPLE
a:= proc(n) options operator, arrow: (1/8)*(2*n-1+(-1)^n)*2^n end proc: seq(a(n), n = 0 .. 30);
MATHEMATICA
LinearRecurrence[{2, 4, -8}, {0, 0, 2}, 31] (* Jean-François Alcover, Nov 16 2022 *)
PROG
(PARI) a(n)=(2*n-1+(-1)^n)*2^n/8 \\ Charles R Greathouse IV, Jul 08 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jul 27 2016
STATUS
approved