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

Author

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

Keywords

• nonn

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).

Programs

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

Magma

R:=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

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 *)

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

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

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