A163214 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

1, 31, 930, 27900, 836535, 25082100, 752044965, 22548807900, 676088221260, 20271372436125, 607803134933490, 18223958540698875, 546414860017738110, 16383333982098029400, 491226816855341457015, 14728612983261055500600

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170750, 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..670
• Index entries for linear recurrences with constant coefficients, signature (29,29,29,-435).

Programs

GAP

a:=[31,930,27900,836535];; for n in [5..20] do a[n]:=29*(a[n-1]+ a[n-2] +a[n-3] -15*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5) )); // G. C. Greubel, Apr 28 2019

Mathematica

coxG[{4,435,-29}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 24 2016 *)
CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(435*t^4-29*t^3-29*t^2 - 29*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{29,29,29,-435}, {1,31, 930,27900,836535}, 20] (* G. C. Greubel, Dec 10 2016 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019

Sage

((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(435*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).

a(n) = 29*(a(n-1) + a(n-2) + a(n-3) - 15*a(n-4)). - G. C. Greubel, Apr 28 2019