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A274497 Sum of the degrees of asymmetry of all binary words of length n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 16:00 EST 2022. Contains 358644 sequences. (Running on oeis4.)