A206799 Based on an erroneous version of A008614.

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

Offset: 0

Author

N. J. A. Sloane, Feb 21 2012

Keywords

• dead

Comments

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

References

• W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955, section 267, page 363

Crossrefs

Cf. A008616.

Programs

Mathematica

(* 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]}]

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

Formula

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))