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

0,3

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

Table of n, a(n) for n=0..30.

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

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 *)

Crossrefs

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

Sequence in context: A032464 A171381 A334083 * A145119 A081411 A269758

Adjacent sequences: A274494 A274495 A274496 * A274498 A274499 A274500

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 27 2016

Status

approved