A274497 Sum of the degrees of asymmetry of all binary words of length n.

0, 0, 2, 4, 16, 32, 96, 192, 512, 1024, 2560, 5120, 12288, 24576, 57344, 114688, 262144, 524288, 1179648, 2359296, 5242880, 10485760, 23068672, 46137344, 100663296, 201326592, 436207616, 872415232, 1879048192, 3758096384, 8053063680

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(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.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,4,-8).

Crossrefs

Cf. A004526, A134353, A274496, A274498, A274499.

Programs

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 *)

PARI

a(n)=(2*n-1+(-1)^n)*2^n/8 \\ Charles R Greathouse IV, Jul 08 2024

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)