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%I #21 Sep 08 2022 08:46:20
%S 1,3,4,6,8,12,16,22,24,34,40,56,62,83,98,133,152,202,236,322,368,496,
%T 570,776,892,1202,1384,1871,2158,2915,3352,4534,5218,7060,8120,10976,
%U 12636,17084,19664,26580,30592,41367,47604,64365,74072,100152,115264,155836,179352,242488,279076,377324,434246,587126
%N Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^7 = 1 >.
%H Colin Barker, <a href="/A298805/b298805.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1).
%F G.f.: (-2*x^18 - 2*x^17 + 3*x^16 + 6*x^15 + 9*x^14 + 12*x^13 + 15*x^12 + 19*x^11 + 21*x^10 + 21*x^9 + 21*x^8 + 21*x^7 + 17*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^16 + x^15 - x^12 - 2*x^11 - x^10 - x^8 - x^6 - 2*x^5 - x^4 + x + 1).
%F The denominator can be factored: G.f. also = -(2*x^18 + 2*x^17 - 3*x^16 - 6*x^15 - 9*x^14 - 12*x^13 - 15*x^12 - 19*x^11 - 21*x^10 - 21*x^9 - 21*x^8 - 21*x^7 - 17*x^6 - 15*x^5 - 13*x^4 - 10*x^3 - 7*x^2 - 4*x - 1) / ((x^4 + x^3 + x^2 + x + 1) * (x^12 - x^10 - x^8 + x^6 - x^4 - x^2 + 1)).
%F a(n) = -a(n-1) + a(n-4) + 2*a(n-5) + a(n-6) + a(n-8) + a(n-10) + 2*a(n-11) + a(n-12) - a(n-15) - a(n-16) for n>18. - _Colin Barker_, Feb 06 2018
%t LinearRecurrence[{-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1},{1,3,4,6,8,12,16,22,24,34,40,56,62,83,98,133,152,202,236},60] (* _Harvey P. Dale_, Jun 15 2021 *)
%o (Magma)
%o // To get the growth function for the group with presentation
%o // < S, T | S^a = T^b = (S*I)^c = 1 >
%o a:=2; b:=3; c:=7;
%o R<x> := RationalFunctionField(Integers());
%o PSR := PowerSeriesRing(Integers():Precision := 100);
%o FG<S,T> := FreeGroup(2);
%o TG := quo<FG | S^a, T^b, (S*T)^c >;
%o f, A :=IsAutomaticGroup(TG);
%o gf := GrowthFunction(A);
%o R!gf;
%o Coefficients(PSR!gf);
%o (PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 17*x^6 + 21*x^7 + 21*x^8 + 21*x^9 + 21*x^10 + 19*x^11 + 15*x^12 + 12*x^13 + 9*x^14 + 6*x^15 + 3*x^16 - 2*x^17 - 2*x^18) / ((1 + x + x^2 + x^3 + x^4)*(1 - x^2 - x^4 + x^6 - x^8 - x^10 + x^12)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018
%Y Cf. A008579, A298802.
%K nonn,easy
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018
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