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A206799
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Based on an erroneous version of A008614.
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0
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4, 1, 0, 2, 4, 3, 4, 4, 4, 5, 4, 6, 8, 7, 8, 8, 8, 9, 12, 10, 12, 15, 12, 12, 16, 17, 16, 18, 20, 19, 20, 20, 24, 25, 24, 26, 28, 27, 28, 32, 32, 33, 36, 34, 36, 39, 40, 40, 44, 45, 44
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OFFSET
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0,1
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COMMENTS
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This is based on the formula in Burnside, Section 267, at the foot of page 363. Unfortunately there is a typo in the formula - the term with numerator 21 should have denominator (1+x)(1-x^3). This produces a sequence with 4's in the denominators. Multiplying by 4 gives a sequence of integers, shown here. This is included in the OEIS in accordance with our policy of publishing incorrect sequences together with pointers to the correct versions. - N. J. A. Sloane, Feb 21 2012
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REFERENCES
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W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955, section 267, page 363
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LINKS
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FORMULA
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A precise definition is: Take the generating function as given by Burnside, expand as a Taylor series, and multiply by 4.
Expansion of (-4 - x + 2 x^3 + x^4 - 2 x^5 - 2 x^6 + 2 x^7 + 3 x^8 + 2 x^9 - 3 x^11)/(-1 + x^3 (1 + x - x^7 - x^8 + x^11))
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MATHEMATICA
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(* expansion*)
w = Exp[I*2*Pi/7];
p[x_] = FullSimplify[ExpandAll[(4/168)*(1/(1 - x)^3 + 21/((1 - x)*(1 - x^2)) + 42/((1 - x)*(1 + x^2)) + 56/(1 - x^3) + 24/((1 - w*x)*(1 - w^2*x)*(1 - w^4*x)) + 24/((1 - w^6*x)*(1 - x*w^5)*(1 - x*w^3)))]];
a = Table[SeriesCoefficient[Series[FullSimplify[ExpandAll[p[x]]], {x, 0, 50}], n], {n, 0, 50}]
(* recursion*)
b[1] = 4; b[2] = 1; b[3] = 0; b[4] = 2; b[5] = 4; b[6] = 3;
b[7] = 4; b[8] = 4; b[9] = 4; b[10] = 5; b[11] = 4;
b[n_Integer?Positive] :=
b[n] = -489 + 11 n + n^2 - b[-11 + n] - 3 b[-10 + n] - 6 b[-9 + n] -
9 b[-8 + n] - 11 b[-7 + n] - 12 b[-6 + n] - 12 b[-5 + n] -
11 b[-4 + n] - 9 b[-3 + n] - 6 b[-2 + n] - 3 b[-1 + n];
Table[b[n], {n, 1, Length[a]}]
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PROG
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(PARI) Vec((-4-x+2*x^3+x^4-2*x^5-2*x^6+2*x^7+3*x^8+2*x^9-3*x^11)/(-1+x^3*(1+x-x^7-x^8+x^11))+O(x^9)) \\ Charles R Greathouse IV, Feb 13 2012
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CROSSREFS
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KEYWORD
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dead
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AUTHOR
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STATUS
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approved
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