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A298802 Growth series for group with presentation < S, T : S^4 = T^4 = (S*T)^4 = 1 >.

%I A298802 #12 Sep 08 2022 08:46:20
%S A298802 1,4,10,24,56,128,294,676,1552,3564,8186,18800,43176,99160,227734,
%T A298802 523020,1201184,2758676,6335658,14550664,33417496,76747632,176260934,
%U A298802 404806196,929690160,2135154556,4903660570,11261895264,25864409480,59400985544,136422101046,313311125788,719559813184
%N A298802 Growth series for group with presentation < S, T : S^4 = T^4 = (S*T)^4 = 1 >.
%H A298802 Colin Barker, <a href="/A298802/b298802.txt">Table of n, a(n) for n = 0..1000</a>
%H A298802 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,-1).
%F A298802 G.f.: (1 + x)^2*(1 + x^2) / (1 - 2*x - 2*x^3 + x^4).
%F A298802 a(n) = 2*a(n-1) + 2*a(n-3) - a(n-4) for n>4. - _Colin Barker_, Feb 04 2018
%t A298802 LinearRecurrence[{2,0,2,-1},{1,4,10,24,56},40] (* _Harvey P. Dale_, Jan 02 2020 *)
%o A298802 (Magma)
%o A298802 R<x> := RationalFunctionField(Integers());
%o A298802 PSR25 := PowerSeriesRing(Integers():Precision := 25);
%o A298802 FG<S,T> := FreeGroup(2);
%o A298802 TG := quo<FG | S^4, T^4, (S*T)^4 >;
%o A298802 f, A :=IsAutomaticGroup(TG);
%o A298802 gf := GrowthFunction(A);
%o A298802 R!gf;
%o A298802 Coefficients(PSR25!gf);
%o A298802 (PARI) Vec((1 + x)^2*(1 + x^2) / (1 - 2*x - 2*x^3 + x^4) + O(x^40)) \\ _Colin Barker_, Feb 04 2018
%Y A298802 Cf. A008579.
%K A298802 nonn,easy
%O A298802 0,2
%A A298802 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018