cp's OEIS Frontend

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

%I A274497 #23 Jul 08 2024 21:44:49
%S A274497 0,0,2,4,16,32,96,192,512,1024,2560,5120,12288,24576,57344,114688,
%T A274497 262144,524288,1179648,2359296,5242880,10485760,23068672,46137344,
%U A274497 100663296,201326592,436207616,872415232,1879048192,3758096384,8053063680
%N A274497 Sum of the degrees of asymmetry of all binary words of length n.
%C A274497 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).
%C A274497 A sequence is palindromic if and only if its degree of asymmetry is 0.
%H A274497 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-8).
%F A274497 a(n) = (1/8)*(2n - 1 + (-1)^n)*2^n.
%F A274497 a(n) = Sum_{k>=0} k*A274496(n,k).
%F A274497 From _Alois P. Heinz_, Jul 27 2016: (Start)
%F A274497 a(n) = 2^(n-1) * A004526(n) = 2^(n-1)*floor(n/2).
%F A274497 a(n) = 2 * A134353(n-2) for n>=2. (End)
%F A274497 From _Chai Wah Wu_, Dec 27 2018: (Start)
%F A274497 a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) for n > 2.
%F A274497 G.f.: 2*x^2/((2*x - 1)^2*(2*x + 1)). (End)
%e A274497 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.
%p A274497 a:= proc(n) options operator, arrow: (1/8)*(2*n-1+(-1)^n)*2^n end proc: seq(a(n), n = 0 .. 30);
%t A274497 LinearRecurrence[{2, 4, -8}, {0, 0, 2}, 31] (* _Jean-François Alcover_, Nov 16 2022 *)
%o A274497 (PARI) a(n)=(2*n-1+(-1)^n)*2^n/8 \\ _Charles R Greathouse IV_, Jul 08 2024
%Y A274497 Cf. A004526, A134353, A274496, A274498, A274499.
%K A274497 nonn,easy
%O A274497 0,3
%A A274497 _Emeric Deutsch_, Jul 27 2016