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A298803 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^4 = 1 >.

%I A298803 #13 Sep 08 2022 08:46:20
%S A298803 1,4,8,16,30,50,88,150,260,448,768,1328,2284,3930,6776,11662,20082,
%T A298803 34592,59560,102570,176642,304180,523830,902084,1553452,2675184,
%U A298803 4606892,7933444,13662066,23527220,40515838,69771678,120152672,206912968,356321478,613615442
%N A298803 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^4 = 1 >.
%H A298803 Colin Barker, <a href="/A298803/b298803.txt">Table of n, a(n) for n = 0..1000</a>
%H A298803 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,3,1,0,-1).
%F A298803 G.f.: (1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)). [Corrected by _Colin Barker_, Feb 04 2018]
%F A298803 a(n) = a(n-2) + 3*a(n-3) + a(n-4) - a(n-6) for n>7. - _Colin Barker_, Feb 04 2018
%t A298803 LinearRecurrence[{0,1,3,1,0,-1},{1,4,8,16,30,50,88,150},40] (* _Harvey P. Dale_, May 03 2019 *)
%o A298803 (Magma)
%o A298803 R<x> := RationalFunctionField(Integers());
%o A298803 PSR25 := PowerSeriesRing(Integers():Precision := 25);
%o A298803 FG<S,T> := FreeGroup(2);
%o A298803 TG := quo<FG | S^3, T^3, (S*T)^4 >;
%o A298803 f, A :=IsAutomaticGroup(TG);
%o A298803 gf := GrowthFunction(A);
%o A298803 R!gf;
%o A298803 Coefficients(PSR25!gf);
%o A298803 (PARI) Vec((1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)) + O(x^40)) \\ _Colin Barker_, Feb 04 2018
%Y A298803 Cf. A008579, A298802.
%K A298803 nonn,easy
%O A298803 0,2
%A A298803 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018