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A289657
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Number of 3-abelian equivalence classes of words of length n over a binary alphabet.
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2
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1, 2, 4, 8, 16, 32, 60, 106, 176, 280, 426, 626, 892, 1238, 1678, 2230, 2910, 3738, 4734, 5920, 7318, 8954, 10852, 13040, 15546, 18400, 21632, 25276, 29364, 33932, 39016, 44654, 50884, 57748, 65286, 73542, 82560, 92386, 103066, 114650, 127186, 140726, 155322, 171028, 187898, 205990, 225360, 246068
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OFFSET
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0,2
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LINKS
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Julien Cassaigne, J. Karhumäki, S. Puzynina, M. A. Whiteland, k-Abelian Equivalence and Rationality, in International Conference on Developments in Language Theory DLT 2016: Developments in Language Theory pp 77-88, Lecture Notes in Computer Science book series (LNCS, volume 9840).
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FORMULA
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See Maple code.
G.f.: (1 - x + x^3 + 2*x^4 + 3*x^5 + x^7 - x^8 + 2*x^9) / ((1 - x)^5*(1 + x)*(1 + x + x^2)). - Colin Barker, Jul 23 2017
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MAPLE
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# This is for n >= 2:
P32:=proc(n) local t1, c, w; c:=exp(Pi*I/3); w:=c^2;
t1:=n^4/18-5*n^3/18+65*n^2/36-23*n/6-(-1)^n/8
+(2/27)*(w^n/c+c/w^n)+1307/216;
round(Re(evalf(simplify(t1))));
end;
[seq(P32(n), n=2..56)];
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MATHEMATICA
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CoefficientList[Series[(1 - x + x^3 + 2 x^4 + 3 x^5 + x^7 - x^8 + 2 x^9)/((1 - x)^5*(1 + x) (1 + x + x^2)), {x, 0, 47}], x] (* Michael De Vlieger, Nov 29 2018 *)
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PROG
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(PARI) Vec((1 - x + x^3 + 2*x^4 + 3*x^5 + x^7 - x^8 + 2*x^9) / ((1 - x)^5*(1 + x)*(1 + x + x^2)) + O(x^50)) \\ Colin Barker, Jul 24 2017
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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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