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A275436 Sum of the asymmetry degrees of all 00-avoiding binary words of length n.

%I A275436 #40 Sep 19 2020 23:20:39
%S A275436 0,0,2,2,8,10,30,40,100,140,320,460,988,1448,2982,4430,8840,13270,
%T A275436 25850,39120,74760,113880,214272,328152,609528,937680,1722890,2660570,
%U A275436 4843400,7503970,13551510,21055480,37759468,58814948,104827712,163642660,290077780
%N A275436 Sum of the asymmetry degrees of all 00-avoiding binary words of length n.
%C A275436 The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
%C A275436 A sequence is palindromic if and only if its asymmetry degree is 0.
%H A275436 Colin Barker, <a href="/A275436/b275436.txt">Table of n, a(n) for n = 0..1000</a>
%H A275436 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,0,-2,-3,1,1).
%F A275436 G.f.: g(z) = 2z^2/((1+z^2)(1+z-z^2)(1-z-z^2)^2).
%F A275436 a(n) = Sum_{k>=0} k * A275435(n,k).
%F A275436 a(n) = ((10*n + 11 + 5*(-1)^n)*F(n) + (10*n - 7 + 15*(-1)^n)*F(n+1) - 4*(sin(n*Pi/2) + 2*cos(n*Pi/2)))/50, where F(n) = A000045(n) (the Fibonacci numbers). - _Emeric Deutsch_, Mar 14 2018
%F A275436 a(n) = a(n-1)+3*a(n-2)-2*a(n-3)-2*a(n-5)-3*a(n-6)+a(n-7)+a(n-8). - _Maxim Karimov_, Aug 23 2020
%e A275436 a(4) = 8 because the 00-avoiding binary words of length 4 are 0101, 0110, 0111, 1010, 1011, 1101, 1110, 1111 and the sum of their asymmetry degrees is 2 + 0 + 1 + 2 + 1 + 1 + 1 + 0 = 8.
%p A275436 g := 2*z^2/((1+z^2)*(1+z-z^2)*(1-z-z^2)^2): gser:= series(g,z = 0,43): seq(coeff(gser, z, n), n = 0 .. 40);
%t A275436 Table[Total@ Map[Total@ BitXor[Take[#, Ceiling[Length[#]/2]], Reverse@ Take[#, -Ceiling[Length[#]/2]]] &, Select[PadLeft[IntegerDigits[#, 2], n] & /@ Range[0, 2^n - 1], Length@ SequenceCases[#, {0, 0}] == 0 &]], {n, 0, 18}] (* _Michael De Vlieger_, Aug 15 2016, Version 10.1 *)
%o A275436 (PARI) concat(vector(2), Vec(2*x^2/((1+x^2)*(1+x-x^2)*(1-x-x^2)^2) + O(x^50))) \\ _Colin Barker_, Aug 29 2016
%Y A275436 Cf. A000045, A275435.
%K A275436 nonn,easy
%O A275436 0,3
%A A275436 _Emeric Deutsch_, Aug 15 2016