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A298812 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^10 = 1 >.

%I A298812 #21 Dec 14 2024 14:36:42
%S A298812 1,3,4,6,8,12,16,24,32,48,62,87,114,165,216,312,408,588,766,1104,1444,
%T A298812 2082,2720,3921,5122,7383,9642,13902,18164,26184,34204,49308,64412,
%U A298812 92856,121298,174867,228438,329313,430188,620160,810132,1167888,1525642,2199372,2873104,4141866,5410628,7799973,10189318,14688939
%N A298812 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^10 = 1 >.
%C A298812 The initial coefficients for the group  S, T : S^2 = T^3 = (S*T)^m = 1 > approach A029744 as m increases.
%H A298812 Colin Barker, <a href="/A298812/b298812.txt">Table of n, a(n) for n = 0..1000</a>
%H A298812 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,0,2,0,4,0,2,0,1,0,0,0,-1).
%F A298812 G.f.: (-2*x^18 + 3*x^16 + 3*x^15 + 6*x^14 + 6*x^13 + 9*x^12 + 9*x^11 + 12*x^10 + 12*x^9 + 12*x^8 + 12*x^7 + 10*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^16 - x^12 - 2*x^10 - 4*x^8 - 2*x^6 - x^4 + 1).
%F A298812 a(n) = a(n-4) + 2*a(n-6) + 4*a(n-8) + 2*a(n-10) + a(n-12) - a(n-16) for n>16. - _Colin Barker_, Feb 06 2018
%o A298812 (Magma) // See Magma program in A298805.
%o A298812 (PARI) Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 10*x^6 + 12*x^7 + 12*x^8 + 12*x^9 + 12*x^10 + 9*x^11 + 9*x^12 + 6*x^13 + 6*x^14 + 3*x^15 + 3*x^16 - 2*x^18) / ((1 + x^2)^2*(1 + x^4)*(1 - 2*x^2 + x^4 - 2*x^6 + x^8)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018
%Y A298812 Cf. A008579, A298802, A298805, A299252, A029744.
%K A298812 nonn,easy
%O A298812 0,2
%A A298812 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018