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}.
Original entry on oeis.org
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
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.
- 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).
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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 *)
A359797
Cogrowth sequence of the lamplighter group Z_2 wr Z where wr denotes the wreath product.
Original entry on oeis.org
1, 3, 15, 87, 547, 3623, 24885, 175591, 1265187, 9271167, 68894785, 518053231, 3935274277, 30158804835, 232930956175, 1811476156847, 14174669041427, 111532445963367, 882004732285473, 7006931317108119, 55899039962599777, 447666261592033123
Offset: 0
Spherical growth sequence for this group is
A288348.
A294683
Growth of the Lamplighter group: number of elements in the Lamplighter group L_2 = Z/2Z wr Z of length up to n with respect to the standard generating set {a,t}.
Original entry on oeis.org
1, 4, 10, 22, 44, 84, 155, 278, 490, 850, 1457, 2474, 4167, 6974, 11609, 19238, 31762, 52274, 85806, 140534, 229735, 374958, 611158, 995016, 1618409, 2630222, 4271663, 6933430, 11248251, 18240668, 29569464, 47920016, 77639264, 125763290, 203680213, 329821130, 534014584
Offset: 0
a(2)=10, since the elements of length up to 2 are 1, a, t, t^-1, at, at^-1, ta, t^2, t^-1a, t^-2.
- Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), 751-759.
- Wikipedia, Lamplighter group
- Index entries for linear recurrences with constant coefficients, signature (1, 3, 0, -5, -3, 2, 3, 1).
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CoefficientList[ Series[((x^2 + x + 1) (x - 1) (x + 1)^3)/((x^3 + x^2 - 1)^2 (x^2 + x - 1)), {x, 0, 36}], x] (* or *)
LinearRecurrence[{1, 3, 0, -5, -3, 2, 3, 1}, {1, 4, 10, 22, 44, 84, 155, 278}, 37] (* Robert G. Wilson v, Aug 08 2018 *)
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Vec((1-x)*(1+x)^3*(1+x+x^2)/((1-x-x^2)*(1-x^2-x^3)^2) + O(x^40)) \\ Michel Marcus, Nov 07 2017
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