A167882 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395622, 172186848, 516560496, 1549681344, 4649043600, 13947129504, 41841384624, 125524142208, 376572391632, 1129717069920

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,-3).

Formula

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*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^16 - 2*t^15 - 2*t^14 - 2*t^13 - 2*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).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = 2*Sum_{j=1..15} a(n-j) - 3*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 3*x + 5*x^16 - 3*x^17). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-3*t+5*t^16-3*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jun 29 2016; Dec 06 2024 *)
coxG[{16, 3, -2}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 06 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-3*x+5*x^16-3*x^17) )); // G. C. Greubel, Dec 06 2024
(SageMath)
def A167882_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-3*x+5*x^16-3*x^17) ).list()
print(A167882_list(40)) # G. C. Greubel, Dec 06 2024
(PARI) Vec((x^16+2*x^15+2*x^14+2*x^13+2*x^12+2*x^11+2*x^10+2*x^9+2*x^8+2*x^7+2*x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(3*x^16-2*x^15-2*x^14-2*x^13-2*x^12-2*x^11-2*x^10-2*x^9-2*x^8-2*x^7-2*x^6-2*x^5-2*x^4-2*x^3-2*x^2-2*x+1)+O(x^99)) \\ Charles R Greathouse IV, May 15 2026

Crossrefs

Cf. A003946, A154638, A169452.

Sequence in context: A166858 A167105 A167649 * A168681 A168729 A168777

Adjacent sequences: A167879 A167880 A167881 * A167883 A167884 A167885

Keyword

nonn,easy

Author

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

Status

approved