A162881 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 43, 1806, 74949, 3109932, 129025155, 5353007478, 222085686501, 9213895794684, 382266301290027, 15859472304395790, 657978118553895573, 27298209939779232636, 1132548704737573481379, 46987204341696557186262

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170762, 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..615
• Index entries for linear recurrences with constant coefficients, signature (41, 41, -861).

Programs

GAP

a:=[43,1806,74949];; for n in [4..20] do a[n]:=41*a[n-1]+41*a[n-2] -861*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(861*t^3-41*t^2-41*t+1))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((x^3+2*x^2+2*x+1)/(861*x^3-41*x^2-41*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

Mathematica

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(861*t^3-41*t^2-41*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 861, -41}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2019 *)

PARI

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(861*t^3-41*t^2-41*t+1)) \\ G. C. Greubel, Oct 24 2018

Sage

((1+x)*(1-x^3)/(1 -42*x +902*x^3 -861*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 27 2019

Formula

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(861*t^3 - 41*t^2 - 41*t + 1).

a(n) = 41*a(n-1) + 41*a(n-2) - 861*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018

G.f.: (1+x)*(1-x^3)/(1 -42*x + 902*x^3 - 861*x^4). - G. C. Greubel, Apr 27 2019