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A165756
Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.
1
1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78726, 236160, 708432, 2125152, 6375024, 19123776, 57367440, 172090656, 516236976, 1548605952, 4645502958, 13935564252, 41803859076, 125403076764, 376183730628, 1128474698076
OFFSET
0,2
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.
FORMULA
G.f.: (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^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).
MAPLE
seq(coeff(series((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Sep 16 2019
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 07 2016 *)
coxG[{10, 3, -2}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 16 2019 *)
PROG
(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11)) \\ G. C. Greubel, Sep 16 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11) )); // G. C. Greubel, Sep 16 2019
(Sage)
def A165756_list(prec):
P.<t> = PowerSeriesRing(ZZ, prec)
return P((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11)).list()
A165756_list(30) # G. C. Greubel, Sep 16 2019
(GAP) a:=[4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78726];; for n in [11..30] do a[n]:=2*Sum([1..9], j-> a[n-j]) -3*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019
CROSSREFS
Sequence in context: A347506 A164697 A165184 * A166328 A166468 A166858
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved