%I #19 Apr 07 2021 05:02:34
%S 1,4,12,36,82,164,294,476,724,1052,1464,1972,2590,3324,4186,5188,6336,
%T 7644,9124,10780,12626,14676,16934,19412,22124,25076,28280,31748,
%U 35486,39508,43826,48444,53376,58636,64228,70164,76458,83116
%N Growth series for Heisenberg group.
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.
%H Moon Duchin, <a href="https://www.ams.org/journals/notices/201608/rnoti-p871.pdf">Counting in Groups: Fine Asymptotic Geometry</a>, Notices of the AMS 63.8 (2016), pp. 871-974. See p. 873. [There may be a typo for c_8 in the recurrence given there]
%H Moon Duchin and Michael Shapiro, <a href="http://arxiv.org/abs/1411.4201">Rational growth in the Heisenberg group</a>, arXiv:1411.4201 [math.GR], 2014; see Section 11.4.2. [There may be a typo in the recurrence given there]
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,5,-6,5,-4,3,-1)
%F G.f.: (1 + x + 4*x^2 + 11*x^3 + 8*x^4 + 21*x^5 + 6*x^6 + 9*x^7 + x^8)/((1-x)^4*(1+x+x^2)*(1+x^2)).
%F a(n) = (c_n + 31*n^3 - 57*n^2 + 105*n)/18 where c_n = -7, -14, 9, -16, -23, 18, -7, -32, 9, 2, -23, 0 for n >= 1, c_{n+12} = c_n. - _R. J. Mathar_, Sep 27 2016
%t LinearRecurrence[{3,-4,5,-6,5,-4,3,-1},{1,4,12,36,82,164,294,476,724},40] (* _Harvey P. Dale_, Sep 02 2018 *)
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Aug 20 2001