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A276061
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Sum of the asymmetry degrees of all compositions of n into parts congruent to 1 mod 3.
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2
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0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 18, 28, 46, 74, 114, 184, 286, 448, 700, 1080, 1676, 2582, 3970, 6104, 9338, 14288, 21808, 33224, 50580, 76844, 116640, 176832, 267740, 405058, 612110, 924204, 1394266, 2101558, 3165406, 4764184, 7165530, 10770386, 16178378
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OFFSET
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0,6
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COMMENTS
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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).
A sequence is palindromic if and only if its asymmetry degree is 0.
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REFERENCES
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S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,0,-1,-1,-1,-2,1,0,1).
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FORMULA
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G.f. g(z) = 2*z^5*(1-z^3)/((1+z)(1-z+z^2)(1+z-z^3)(1-z-z^3)^2). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))(1-F(z))^2).
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EXAMPLE
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a(7) = 4 because the compositions of 7 with parts in {1,4,7,10,... } are 7, 4111, 1411, 1141, 1114, and 1111111, and the sum of their asymmetry degrees is 0+1+1+1+1+0.
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MAPLE
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g := 2*z^5*(1-z^3)/((1+z)*(1-z+z^2)*(1+z-z^3)*(1-z-z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
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MATHEMATICA
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Table[Total@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; Mod[a, 3] != 1]], 1]]], {n, 0, 36}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
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PROG
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(PARI) concat(vector(5), Vec(2*x^5*(1-x^3)/((1+x)*(1-x+x^2)*(1+x-x^3)*(1-x-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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