A169400 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^32 = I.

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset

0,2

Comments

The initial terms coincide with those of A003945, although the two sequences are eventually different.

First disagreement is at index 32, the difference is 3. - Klaus Brockhaus, Jun 26 2011

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1).

Formula

G.f.: (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)/(t^32 - t^31 - t^30 - t^29 - t^28 - t^27 - t^26 - t^25 - t^24 - t^23 - t^22 - t^21 - t^20 - t^19 - t^18 - t^17 - t^16 - t^15 - t^14 - t^13 - t^12 - t^11 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2 - t + 1).

G.f.: (1+2*sum(k=1..31,x^k)+x^32)/(1-sum(k=1..31,x^k)+x^32).

Maple

(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)/(t^32 - t^31 - t^30 - t^29 - t^28 - t^27 - t^26 - t^25 - t^24 - t^23 - t^22 - t^21 - t^20 - t^19 - t^18 - t^17 - t^16 - t^15 - t^14 - t^13 - t^12 - t^11 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2 - t + 1) ;
taylor(%, t=0, 65) ;
gfun[seriestolist](%) ; # R. J. Mathar, Apr 12 2019

Mathematica

(1 + 2*Sum[x^k, {k, 1, 31}] + x^32)/(1 - Sum[x^k, {k, 1, 31}] + x^32) + O[x]^32 // CoefficientList[#, x]& (* Jean-François Alcover, Apr 12 2019 *)
coxG[{32, 32, -1, 40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 17 2022 *)

Prog

(PARI) my(x='x+O('x^66)); Vec((1+2*sum(k=1, 31, x^k)+x^32)/(1-sum(k=1, 31, x^k)+x^32)) /* Joerg Arndt, Jun 26 2011 */

Crossrefs

Cf. A003945 (G.f.: (1+x)/(1-2*x) ).

Sequence in context: A169256 A169304 A169352 * A169448 A169496 A169544

Adjacent sequences: A169397 A169398 A169399 * A169401 A169402 A169403

Keyword

nonn

Author

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

Extensions

Deleted suspicious link to Index and a suspect b-file. - N. J. A. Sloane, Apr 12 2019

Status

approved