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A301730 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).

%I A301730 #15 Sep 08 2022 08:46:20
%S A301730 1,6,11,14,18,24,30,34,38,42,48,54,58,62,66,72,78,82,86,90,96,102,106,
%T A301730 110,114,120,126,130,134,138,144,150,154,158,162,168,174,178,182,186,
%U A301730 192,198,202,206,210,216,222,226,230,234,240,246,250,254,258,264
%N A301730 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 A301730 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 A301730 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 A301730 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 A301730 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A301730 From _Bruno Berselli_, Apr 09 2018: (Start)
%F A301730 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 A301730 a(5*k)   = 24*k with k>0, a(0)=1;
%F A301730 a(5*k+1) = 24*k + 6;
%F A301730 a(5*k+2) = 24*k + 10 with k>0, a(2)=11;
%F A301730 a(5*k+3) = 24*k + 14;
%F A301730 a(5*k+4) = 24*k + 18. (End)
%o A301730 (Magma)
%o A301730 R<x> := RationalFunctionField(Integers());
%o A301730 FG3<X,Y,Z> := FreeGroup(3);
%o A301730 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 A301730 G3 := AutomaticGroup(Q3);
%o A301730 f3 := GrowthFunction(G3);
%o A301730 R!f3;
%o A301730 PSR := PowerSeriesRing(Integers():Precision := 60);
%o A301730 Coefficients(PSR!f3);
%Y A301730 Cf. A072154.
%K A301730 nonn,easy
%O A301730 0,2
%A A301730 _N. J. A. Sloane_, Mar 30 2018