A063810 Growth series for Heisenberg group.

1, 4, 12, 36, 82, 164, 294, 476, 724, 1052, 1464, 1972, 2590, 3324, 4186, 5188, 6336, 7644, 9124, 10780, 12626, 14676, 16934, 19412, 22124, 25076, 28280, 31748, 35486, 39508, 43826, 48444, 53376, 58636, 64228, 70164, 76458, 83116

Offset

0,2

References

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

Links

Table of n, a(n) for n=0..37.

Moon Duchin, Counting in Groups: Fine Asymptotic Geometry, 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]

Moon Duchin and Michael Shapiro, Rational growth in the Heisenberg group, arXiv:1411.4201 [math.GR], 2014; see Section 11.4.2. [There may be a typo in the recurrence given there]

Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-6,5,-4,3,-1)

Formula

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)).

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

Mathematica

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 *)

Crossrefs

Sequence in context: A307182 A357061 A190072 * A183931 A320967 A261584

Adjacent sequences: A063807 A063808 A063809 * A063811 A063812 A063813

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 20 2001

Status

approved