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Expansion of (x^8-x^7+x^6+5*x^5+4*x^4+3*x^3+5*x^2+5*x+1)/(x^6-x^5-x+1).
1

%I #15 Sep 08 2022 08:46:20

%S 1,6,11,14,18,24,30,34,38,42,48,54,58,62,66,72,78,82,86,90,96,102,106,

%T 110,114,120,126,130,134,138,144,150,154,158,162,168,174,178,182,186,

%U 192,198,202,206,210,216,222,226,230,234,240,246,250,254,258,264

%N Expansion of (x^8-x^7+x^6+5*x^5+4*x^4+3*x^3+5*x^2+5*x+1)/(x^6-x^5-x+1).

%C Growth series for group with presentation < X, Y, Z | X^2 = Y^2, X^2 = Z^2, X^2 = (Y*Z)^3, X^2 = (Z*X)^2, X^2 = (X*Y)^6 >. Probably Shutov intended to add "X^2 = Id" to the presentation, which would have produced the sequence A072154.

%H A. V. Shutov, <a href="http://mi.mathnet.ru/eng/znsl933">On the number of words of a given length in plane crystallographic groups (Russian)</a>, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203. See Table 1, line "p6m".

%H A. V. Shutov, <a href="https://doi.org/10.1007/s10958-005-0329-2">On the number of words of a given length in plane crystallographic groups (English translation)</a>, J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m".

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F From _Bruno Berselli_, Apr 09 2018: (Start)

%F G.f.: (x + 1)*(x^7 - 2*x^6 + 3*x^5 + 2*x^4 + 2*x^3 + x^2 + 4*x + 1)/((x - 1)^2*(x^4 + x^3 + x^2 + x + 1)).

%F a(5*k) = 24*k with k>0, a(0)=1;

%F a(5*k+1) = 24*k + 6;

%F a(5*k+2) = 24*k + 10 with k>0, a(2)=11;

%F a(5*k+3) = 24*k + 14;

%F a(5*k+4) = 24*k + 18. (End)

%o (Magma)

%o R<x> := RationalFunctionField(Integers());

%o FG3<X,Y,Z> := FreeGroup(3);

%o Q3 := quo<FG3| X^2=Y^2, X^2=Z^2, X^2 = (Y*Z)^3, X^2 = (Z*X)^2, X^2 = (X*Y)^6 >;

%o G3 := AutomaticGroup(Q3);

%o f3 := GrowthFunction(G3);

%o R!f3;

%o PSR := PowerSeriesRing(Integers():Precision := 60);

%o Coefficients(PSR!f3);

%Y Cf. A072154.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 30 2018