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A058766
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a(0) = 1, a(1) = 2; for n>=2 a(n) is the number of degree-n reducible polynomials over GF(2).
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9
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1, 2, 3, 6, 13, 26, 55, 110, 226, 456, 925, 1862, 3761, 7562, 15223, 30586, 61456, 123362, 247612, 496694, 996199, 1997294, 4003747, 8023886, 16078346, 32212256, 64528069, 129246720, 258849061, 518358122, 1037951557, 2078209982, 4160751616
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OFFSET
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0,2
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COMMENTS
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Dimensions of homogeneous subspaces of shuffle algebra defined in the "Comments" line.
Let x and y be two letters, m and m' any two words, e is the empty word of the free monoid generated by (x,y). Let uu denote the shuffle or Hurwitz product: xm uu ym' =x.(m uu ym') + y.(xm uu m'); of course, e is neutral.
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REFERENCES
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M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).
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LINKS
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FORMULA
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EXAMPLE
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Degree 3: x uu x = 2 x^2, y uu y = 2 y^2, x uu y = xy + yx.
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MATHEMATICA
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a[n_] := 2^n - DivisorSum[n, MoebiusMu[n/#] * 2^# &] / n; a[0] = 1; a[1] = 2; Array[a, 33, 0] (* Amiram Eldar, Aug 13 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Claude Lenormand (claude.lenormand(AT)free.fr), Jan 03 2001
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EXTENSIONS
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Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002
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STATUS
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approved
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