A166468 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708582, 2125728, 6377136, 19131264, 57393360, 172178784, 516532464, 1549585728, 4648722192, 13946061600, 41837869872, 125512664832, 376535160174, 1129596977628

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A003946, although the two sequences are eventually different.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -3).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13) )); // G. C. Greubel, Apr 26 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13), {x, 0, 30}], x ] (* Vincenzo Librandi, Apr 29 2014 *)(* modified by G. C. Greubel, Apr 26 2019 *)
coxG[{12,3,-2,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 09 2018 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)) \\ G. C. Greubel, Apr 26 2019

Sage

((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

Formula

G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^12 - 2*t^11 - 2*t^10 - 2*t^9 - 2*t^8 - 2*t^7 - 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).

G.f.: (1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13). - G. C. Greubel, Apr 26 2019

a(n) = -3*a(n-12) + 2*Sum_{k=1..11} a(n-k). - Wesley Ivan Hurt, May 06 2021