A275436 Sum of the asymmetry degrees of all 00-avoiding binary words of length n.
0, 0, 2, 2, 8, 10, 30, 40, 100, 140, 320, 460, 988, 1448, 2982, 4430, 8840, 13270, 25850, 39120, 74760, 113880, 214272, 328152, 609528, 937680, 1722890, 2660570, 4843400, 7503970, 13551510, 21055480, 37759468, 58814948, 104827712, 163642660, 290077780
Offset: 0
Examples
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.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,3,-2,0,-2,-3,1,1).
Programs
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Maple
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);
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Mathematica
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 *) -
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
Formula
G.f.: g(z) = 2z^2/((1+z^2)(1+z-z^2)(1-z-z^2)^2).
a(n) = Sum_{k>=0} k * A275435(n,k).
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
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
Comments