A294782 Spherical growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length n with respect to the standard generating set {a,t}.
1, 4, 12, 36, 100, 268, 704, 1812, 4600, 11556, 28788, 71252, 175452, 430284, 1051848, 2564708, 6240752, 15161092, 36784284, 89155268, 215911636, 522543436, 1263991824, 3056244212, 7387384808, 17851786148, 43130479748, 104187860340, 251648811212, 607755975820, 1467673342616
Offset: 0
Keywords
Examples
a(2)=12, since the elements of length 2 are a^2, at, at^-1, a^-2, a^-1t, a^-1t^-1, ta, ta^-1, t^2, t^-1a, t^-1a^-1, t^-2.
Links
- Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), 751-759.
- Index entries for linear recurrences with constant coefficients, signature (4, -2, -4, -4, 4, 6, 4, 1).
Programs
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Mathematica
LinearRecurrence[{4,-2,-4,-4,4,6,4,1},{1,4,12,36,100,268,704,1812,4600},40] (* Harvey P. Dale, Jan 31 2025 *)
Formula
G.f.: (1-x)^3 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).
Comments