A154638 -id:A154638 - cp's OEIS frontend

A169452 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^33 = I.

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892032, 19747769352192, 118486616113152, 710919696678912, 4265518180073472

Offset: 0

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

The initial terms coincide with those of A003949, although the two sequences are eventually different.

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4) )); // G. C. Greubel, May 01 2019

gf:= (t+1) *(t^2+t+1) *(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1) *(t^20-t^19+t^17-t^16 +t^14-t^13+t^11-t^10+t^9-t^7+t^6-t^4+t^3- t+1) / (15*t^33-5*t^32-5*t^31-5*t^30-5*t^29 -5*t^28-5*t^27 -5*t^26-5*t^25 -5*t^24 -5*t^23-5*t^22-5*t^21-5*t^20 -5*t^19-5*t^18-5*t^17 -5*t^16 -5*t^15 -5*t^14-5*t^13-5*t^12-5*t^11-5*t^10-5*t^9-5*t^8-5*t^7 -5*t^6 -5*t^5-5*t^4 -5*t^3-5*t^2-5*t+1):
S:= series(gf,t,101):
seq(coeff(S,t,j),j=0..100); # Robert Israel, Aug 26 2014

coxG[{pwr_,c1_,c2_,trms_:20}]:=Module[{num=Total[2t^Range[pwr-1]]+t^pwr+ 1, den =Total[c2*t^Range[pwr-1]]+c1*t^pwr+1},CoefficientList[ Series[ num/den,{t,0,trms}],t]]; coxG[{33,15,-5,30}]
(* "pwr" is the largest exponent in the g.f.;
"c1" is the first coefficient in the denominator of the g.f.;
"c2" is the second coefficient in the denominator of the g.f.;
"trms" is the number of terms desired (with a default number of 20) *)
(* Harvey P. Dale, Aug 16 2014 *)
CoefficientList[Series[(1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34), {x,0,25}], x] (* G. C. Greubel, May 01 2019 *)

my(x='x+O('x^25)); Vec((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)) \\ G. C. Greubel, May 01 2019

((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)).series(x, 25).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

G.f.: (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)/(15*t^33 - 5*t^32 - 5*t^31 - 5*t^30 - 5*t^29 - 5*t^28 - 5*t^27 - 5*t^26 - 5*t^25 - 5*t^24 - 5*t^23 - 5*t^22 - 5*t^21 - 5*t^20 - 5*t^19 - 5*t^18 - 5*t^17 - 5*t^16 - 5*t^15 - 5*t^14 - 5*t^13 - 5*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).
G.f.: (1+x)*(1-x^33)/(1 - 6*x + 20*x^33 - 15*x^34). - G. C. Greubel, May 01 2019
a(n) = -15*a(n-33) + 5*Sum_{k=1..32} a(n-k). - Wesley Ivan Hurt, May 06 2021

A161409 Number of reduced words of length n in the Weyl group E_6 on 6 generators and order 51840.

Original entry on oeis.org

1, 6, 20, 50, 105, 195, 329, 514, 754, 1048, 1389, 1765, 2159, 2549, 2911, 3222, 3461, 3611, 3662, 3611, 3461, 3222, 2911, 2549, 2159, 1765, 1389, 1048, 754, 514, 329, 195, 105, 50, 20, 6, 1

Offset: 0

John Cannon and N. J. A. Sloane, Nov 29 2009

			Coxeter matrix:
. [1 2 3 2 2 2]
. [2 1 2 3 2 2]
. [3 2 1 3 2 2]
. [2 3 3 1 3 2]
. [2 2 2 3 1 3]
. [2 2 2 2 3 1]

N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Index entries for sequences related to groups

Cf. A161410, A154638.

G := CoxeterGroup(GrpFPCox, "E6");
f := GrowthFunction(G);
Coefficients(PolynomialRing(IntegerRing())!f);
// Corrected by Klaus Brockhaus, Feb 12 2010

CoefficientList[Series[((1-x^2) (1-x^5) (1-x^6) (1-x^8) (1-x^9) (1-x^12))/(1-x)^6,{x,0,40}],x] (* Harvey P. Dale, Aug 17 2011 *)

G.f.: f(2)f(5)f(6)f(8)f(9)f(12)/f(1)^6 where f(k) = 1-x^k.

A163876 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 93, 180, 351, 684, 1332, 2592, 5046, 9825, 19128, 37239, 72498, 141144, 274788, 534972, 1041513, 2027676, 3947595, 7685400, 14962368, 29129580, 56711106, 110408373, 214949232, 418475259, 814711182, 1586125572, 3087958512

Offset: 0

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

Also, coordination sequence for (6,6,6) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The initial terms coincide with those of A003945, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, -1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7) )); // G. C. Greubel, Apr 25 2019

coxG[{6,1,-1,40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 22 2015 *)
CoefficientList[Series[(1+x)*(1-x^6)/(1-2*x+2*x^6-x^7), {x,0,40}], x] (* G. C. Greubel, Aug 06 2017, modified Apr 25 2019 *)

x='x+O('x^40); Vec((x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(x^6-x^5- x^4-x^3-x^2-x+1)) \\ G. C. Greubel, Aug 06 2017

((1+x)*(1-x^6)/(1-2*x+2*x^6-x^7)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(x^6 - x^5 - x^4 - x^3 - x^2 - x + 1).
G.f.: (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7). - G. C. Greubel, Apr 25 2019
a(n) = -a(n-6) + Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 07 2021

A154645 a(n) is the ratio of the sum of the squares of the bends of the spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, using "strategy (b)" to count them (see the reference), to the sum of the squares of the bends of the initial five mutually tangent spheres.

Original entry on oeis.org

1, 6, 77, 1278, 15978, 216366

Offset: 0

Colin Mallows, Jan 13 2009

In strategy (b) we count all spheres that can be generated (by reflection) from all quintuples that appeared in the previous generation.

			Starting with five spheres with bends 0,0,1,1,1, the first derived generation has 5 spheres with bends 1,1,1,3,3, so a(2) = 9/3 = 3.

C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.

For other sequences relating to the 3-dimensional case, see A154638-A154645.

A189226 Curvatures in the nickel-dime-quarter Apollonian circle packing, ordered first by generation and then by size.

Original entry on oeis.org

-11, 21, 24, 28, 40, 52, 61, 157, 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397, 132, 156, 160, 189, 204, 205, 216, 237, 253, 285, 288, 309, 316, 336, 349, 405, 412, 421, 453, 460, 469, 472, 517, 544, 565, 616, 628, 685, 717, 741, 1084, 1093, 1104, 1125, 1128, 1140

Offset: 1

Jonathan Sondow, Apr 18 2011

For a circle, curvature = 1/radius. The curvatures of a quarter, nickel, and dime are approximately proportional to 21, 24, and 28, respectively. Three mutually tangent circles with curvatures 21, 24, 28 can be inscribed in a circle of curvature 11.
Apollonius's and Descartes's Theorems say that, given three mutually tangent circles of curvatures a, b, c, there are exactly two circles tangent to all three, and their curvatures are a + b + c +- 2*sqrt(ab + ac + bc). (Here negative curvature of one of the two circles means that the three circles are inscribed in it.)
Fuchs (2009) says "An Apollonian circle packing ... is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well." That is because if a + b + c - 2s*qrt(ab + ac + bc) is an integer, then so is a + b + c + 2*sqrt(ab + ac + bc).
For n > 1, the n-th generation of the packing has 4*3^(n-2) circles.
Infinitely many of the curvatures are prime numbers A189227. In fact, in any integral Apollonian circle packing that is primitive (i.e., the curvatures have no common factor), the prime curvatures constitute a positive fraction of all primes (Bourgain 2012) and there are infinitely many pairs of tangent circles both of whose curvatures are prime (Sarnak 2007, 2011).
Fuchs and Sanden (2012) report on experiments with the nickel-dime-quarter Apollonian circle packing, which they call the coins packing P_C.

			The 1st-generation curvatures are -11, 21, 24, 28, the 2nd are 40, 52, 61, 157, and the 3rd are 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397. The 4th generation begins 132, 156, 160, 189, 204, 205, 216, ....
As 21 + 24 + 28 +- 2*sqrt(21*24 + 21*28 + 24*28) = 157 or -11, the sequence begins -11, 21, 24, 28, ... and 157 is in it.
The primes 157 and 397 are the curvatures of two circles that are tangent.

D. Austin, When Kissing Involves Trigonometry, AMS feature column March 2006.
J. Bourgain, Integral Apollonian circle packings and prime curvatures, arXiv:1105.5127 [math.NT], 2011-2012.
J. Bourgain and A. Kontorovich, On the Strong Density Conjecture for Integral Apollonian Circle Packings, arXiv:1205.4416 [math.NT], 2012-2013. See figure 1.
S. Butler, R. Graham, G. Guettler and C. Mallows, Irreducible Apollonian configurations and packings, Discrete & Computational Geometry, 44 (2010), 487-507.
E. Fuchs, Arithmetic Properties of Apollonian Circle Packings, Ph.D. thesis 2009.
E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Experiment. Math. 20 (2011), 380-399.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group., Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.
K. E. Hirst, The Apollonian Packing of Circles, J. London Math. Soc. s1-42(1) (1967), 281-291.
E. Kasner and F. Supnick, The Apollonian packing of circles, Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 378-384.
A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc., 50 (2013), 187-228.
J. C. Lagarias, C. L. Mallows, and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
L. Levine, W. Pegden, C. K. Smart, Apollonian Structure in the Abelian Sandpile, arXiv:1208.4839 [math.AP], 2012-2014.
D. Mackenzie, A Tisket, a Tasket, an Apollonian Gasket, American Scientist, 98 (2010).
C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.
I. Peterson, Circle game, Science News, 4/21/01.
I. Peterson, Temple circles, Math Trek, 4/23/01.
P. Sarnak, Letter to Lagarias on integral Apollonian packings, June, 2007.
P. Sarnak, Integral Apollonian packings, MAA Lecture, Jan 2009.
P. Sarnak, Integral Apollonian packings, Amer. Math. Monthly, 118 (2011), 291-306.
K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014.
Wikipedia, Integral Apollonian circle packings
Wikipedia, Descartes' theorem

Cf. A042944, A042946, A042945, A045506, A045673, A045864, A052483, A060790, A135849, A137246, A154636, A154637, A154638, A171090, A189227, A218155, A248938, A265495, A294510.

root = {-11, 21, 24, 28};
triples = Subsets[root, {3}];
a = {root};
Do[
  ng = Table[Total@t + 2 Sqrt@Total[Times @@@ Subsets[t, {2}]], {t, triples}];
  AppendTo[a, Sort@ng];
  triples = Join @@ Table[{t, r} = tr; Table[Append[p, r], {p, Subsets[t, {2}]}], {tr, Transpose@{triples, ng}}]
  , {k, 3}];
Flatten@a (* Andrey Zabolotskiy, May 29 2022 *)

a(n) == 0, 4, 12, 13, 16, or 21 (mod 24).

Terms a(28) and beyond from Andrey Zabolotskiy, May 29 2022

A167931 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 20, 380, 7220, 137180, 2606420, 49521980, 940917620, 17877434780, 339671260820, 6453753955580, 122621325156020, 2329805177964380, 44266298381323220, 841059669245141180, 15980133715657682420

Offset: 0

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

nonn

The initial terms coincide with those of A170739, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, -171).

Cf. A154638, A170739.

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17) ));  // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17), {x, 0, 20}], x] (* G. C. Greubel, Jul 01 2016, modified Apr 25 2019 *)
coxG[{16, 171, -18, 20}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

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)/( 171*t^16 - 18*t^15 - 18*t^14 - 18*t^13 - 18*t^12 - 18*t^11 - 18*t^10 - 18*t^9 - 18*t^8 - 18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 - 18*t + 1).

G.f.: (1+x)*(1-x^16)/(1 -19*x +189*x^16 -171*x^17). - G. C. Greubel, Apr 25 2019

A167933 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10752000000000, 215040000000000, 4300800000000000, 86016000000000000, 1720320000000000000, 34406400000000000000

Offset: 0

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

nonn

The initial terms coincide with those of A170740, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, -190).

Cf. A154638, A170740.

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17), {x, 0, 20}], x] (* G. C. Greubel, Jul 01 2016, modified Apr 25 2019 *)
coxG[{16, 190, -19, 20}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

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)/( 190*t^16 - 19*t^15 - 19*t^14 - 19*t^13 - 19*t^12 - 19*t^11 - 19*t^10 - 19*t^9 - 19*t^8 - 19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

G.f.: (1+x)*(1-x^16)/(1 -20*x +209*x^16 -190*x^17). - G. C. Greubel, Apr 25 2019

A167881 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98301, 196596, 393183, 786348, 1572660, 3145248, 6290352, 12580416, 25160256, 50319360, 100636416, 201268224, 402527232, 805036032, 1610035200, 3219996672

Offset: 0

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

nonn

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

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

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

Cf. A003945, A154638, A169452.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-2*x+2*x^16-x^17) )); // G. C. Greubel, Dec 06 2024

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

def A167881_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-2*x+2*x^16-x^17) ).list()
print(A167881_list(40)) # G. C. Greubel, Dec 06 2024

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)/(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).
From G. C. Greubel, Dec 06 2024: (Start)
a(n) = Sum_{j=1..15} a(n-j) - a(n-16).
G.f.: (1+x)*(1-x^16)/(1 - 2*x + 2*x^16 - x^17). (End)

A167929 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516992, 67838877305856, 1221099791505408, 21979796247097344, 395636332447752192, 7121453984059539456

Offset: 0

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

nonn

The initial terms coincide with those of A170738, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, -153).

Cf. A154638, A170738.

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17), {x, 0, 20}], x] (* G. C. Greubel, Jul 01 2016, modified Apr 26 2019 *)
coxG[{16, 153, -17}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

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)/( 153*t^16 - 17*t^15 - 17*t^14 - 17*t^13 - 17*t^12 - 17*t^11 - 17*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).
G.f.: (1+x)*(1-x^16)/(1 - 18*x + 170*x^16 - 153*x^17). - G. C. Greubel, Apr 26 2019
a(n) = -153*a(n-16) + 17*Sum_{k=1..15} a(n-k). - Wesley Ivan Hurt, May 06 2021

A167935 Number of reduced words of length n in Coxeter group on 22 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 22, 462, 9702, 203742, 4278582, 89850222, 1886854662, 39623947902, 832102905942, 17474161024782, 366957381520422, 7706105011928862, 161828205250506102, 3398392310260628142, 71366238515473190982

Offset: 0

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

nonn

The initial terms coincide with those of A170741, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, -210).

Cf. A154638, A170741.

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-21*x+230*x^16-210*x^17) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^16)/(1-21*x+230*x^16-210*x^17), {x, 0, 20}], x] (* G. C. Greubel, Jul 01 2016, modified Apr 26 2019 *)
coxG[{16, 210, -20}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-21*x+230*x^16-210*x^17)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^16)/(1-21*x+230*x^16-210*x^17)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

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)/(210*t^16 - 20*t^15 - 20*t^14 - 20*t^13 - 20*t^12 - 20*t^11 - 20*t^10 - 20*t^9 - 20*t^8 - 20*t^7 - 20*t^6 - 20*t^5 - 20*t^4 - 20*t^3 - 20*t^2 - 20*t + 1).
G.f.: (1+x)*(1-x^16)/(1 - 21*x + 230*x^16 - 210*x^17). - G. C. Greubel, Apr 26 2019
a(n) = -210*a(n-16) + 20*Sum_{k=1..15} a(n-k). - Wesley Ivan Hurt, May 06 2021

cp's OEIS Frontend

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A167931 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A167933 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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