A163314 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 3, 6, 12, 24, 45, 84, 159, 300, 564, 1062, 2001, 3768, 7095, 13362, 25164, 47388, 89241, 168060, 316491, 596016, 1122420, 2113746, 3980613, 7496304, 14117067, 26585310, 50065548, 94283616, 177555237, 334372644, 629691735, 1185837684

Offset

0,2

Comments

The initial terms coincide with those of A003945, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. O'Keeffe, Coordination sequences for hyperbolic tilings, Zeitschrift für Kristallographie, 213 (1998), 135-140 (see next-to-last table, row {10, 3}).

Index entries for linear recurrences with constant coefficients, signature (2, -1, 2, -1).

Formula

G.f.: (t^4 + t^3 + t^2 + t + 1)/(t^4 - 2*t^3 + t^2 - 2*t + 1).

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-a(n-4). - Wesley Ivan Hurt, May 10 2021

Mathematica

CoefficientList[Series[(t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1), {t, 0, 40} ], t] (* or *) LinearRecurrence[{2, -1, 2, -1}, {1, 3, 6, 12, 24}, 40] (* G. C. Greubel, Dec 18 2016 *)

Prog

(PARI) my(t='t+O('t^40)); Vec((t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1)) \\ G. C. Greubel, Dec 18 2016
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1) )); // G. C. Greubel, May 12 2019
(Sage) ((x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019
(GAP) a:=[3, 6, 12, 24];; for n in [5..40] do a[n]:=2*a[n-1]-a[n-2]+ 2*a[n-3]-a[n-4]; od; Concatenation([1], a); # G. C. Greubel, May 12 2019

Crossrefs

Sequence in context: A364497 A132974 A132979 * A018183 A196787 A367217

Adjacent sequences: A163311 A163312 A163313 * A163315 A163316 A163317

Keyword

nonn

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Status

approved