A170017 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^36 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079544, 265863444556808, 1861044111897656, 13027308783283592

Offset

0,2

Comments

The initial terms coincide with those of A003950, although the two sequences are eventually different.

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

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -21).

Formula

G.f.: (t^36 + 2*t^35 + 2*t^34 + 2*t^33 + 2*t^32 + 2*t^31 + 2*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)/(21*t^36 - 6*t^35 - 6*t^34 - 6*t^33 - 6*t^32

- 6*t^31 - 6*t^30 - 6*t^29 - 6*t^28 - 6*t^27 - 6*t^26 - 6*t^25 - 6*t^24

- 6*t^23 - 6*t^22 - 6*t^21 - 6*t^20 - 6*t^19 - 6*t^18 - 6*t^17 - 6*t^16

- 6*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 -

6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-36) + 6*Sum_{k=1..35} a(n-k). - Wesley Ivan Hurt, May 25 2024

Mathematica

With[{num=Total[2t^Range[35]]+t^36+1, den=Total[-6 t^Range[ 35]]+ 21t^36+ 1}, CoefficientList[Series[num/den, {t, 0, 50}], t]] (* Harvey P. Dale, Jun 20 2012 *)

Crossrefs

Sequence in context: A169453 A169501 A169549 * A170065 A170113 A170161

Adjacent sequences: A170014 A170015 A170016 * A170018 A170019 A170020

Keyword

nonn

Author

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

Status

approved