A169305 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^30 = I.

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset

0,2

Comments

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

First disagreement at index 30: a(30) = 274521509459526, A003946(30) = 274521509459532. - Klaus Brockhaus, Jun 22 2011

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

Links

Table of n, a(n) for n=0..25.

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

Formula

G.f.: (t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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^30 - 2*t^29 - 2*t^28 - 2*t^27 - 2*t^26 - 2*t^25 - 2*t^24 - 2*t^23 - 2*t^22 - 2*t^21 - 2*t^20 - 2*t^19 - 2*t^18 - 2*t^17 - 2*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).

a(n) = -3*a(n-30) + 2*Sum_{k=1..29} a(n-k). - Wesley Ivan Hurt, Dec 31 2023

Mathematica

coxG[{30, 3, -2, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 18 2022 *)

Crossrefs

Cf. A003946 (G.f.: (1+x)/(1-3*x)).

Sequence in context: A169161 A169209 A169257 * A169353 A169401 A169449

Adjacent sequences: A169302 A169303 A169304 * A169306 A169307 A169308

Keyword

nonn

Author

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

Status

approved