A154638 -id:A154638 - cp's OEIS frontend

A164779 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829645, 430466400, 3874194000, 34867713600, 313809130800, 2824279552800, 25418492355600, 228766218624000, 2058894054430380, 18530029271219040, 166770108473225760

Offset: 0

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

nonn

The initial terms coincide with those of A003952, 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
Index entries for linear recurrences with constant coefficients, signature (8, 8, 8, 8, 8, 8, 8, -36).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^8)/(1-9*x+44*x^8-36*x^9) )); // G. C. Greubel, Apr 26 2019

coxG[{8,36,-8}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 13 2017 *)
CoefficientList[Series[(1+x)*(1-x^8)/(1-9*x+44*x^8-36*x^9), {x,0,20}], x] (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^8)/(1-9*x+44*x^8-36*x^9)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^8)/(1-9*x+44*x^8-36*x^9)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 36*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).

G.f.: (1+x)*(1-x^8)/(1 -9*x +44*x^8 -36*x^9). - G. C. Greubel, Apr 26 2019

A165699 Number of reduced words of length n in Coxeter group on 45 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

Original entry on oeis.org

1, 45, 1980, 87120, 3833280, 168664320, 7421230080, 326534123520, 14367501434880, 632170063133730, 27815482777840560, 1223881242223068990, 53850774657730746960, 2369434084936444167840, 104255099737040360655360

Offset: 0

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

nonn

The initial terms coincide with those of A170764, 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..600
Index entries for linear recurrences with constant coefficients, signature (43, 43, 43, 43, 43, 43, 43, 43, -946).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^9)/(1-44*x+989*x^9-946*x^10) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^9)/(1-44*x+989*x^9-946*x^10), {x, 0, 20}], x] (* or *) coxG[{9, 946, -43}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^9)/(1-44*x+989*x^9-946*x^10)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^9)/(1-44*x+989*x^9-946*x^10)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (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)/(946*t^9 - 43*t^8 - 43*t^7 - 43*t^6 - 43*t^5 - 43*t^4 - 43*t^3 - 43*t^2 - 43*t + 1).

G.f.: (1+x)*(1-x^9)/(1 -44*x +989*x^9 -946*x^10). - G. C. Greubel, Apr 26 2019

A166377 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144576, 804925734834, 9659108817072, 115909305793710, 1390911669390672, 16690940031081888, 200291280353708544

Offset: 0

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

nonn

The initial terms coincide with those of A170732, 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 (11, 11, 11, 11, 11, 11, 11, 11, 11, 11, -66).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^11)/(1-12*x+77*x^11-66*x^12) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+t)*(1-t^11)/(1-12*t+77*t^11-66*t^12), {t, 0, 20}], t] (* G. C. Greubel, May 10 2016 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^11)/(1-12*x+77*x^11-66*x^12)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^11)/(1-12*x+77*x^11-66*x^12)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (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)/(66*t^11 - 11*t^10 - 11*t^9 - 11*t^8 - 11*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).

G.f.: (1 + t - t^11 - t^12)/(1 - 12*t + 77*t^11 - 66*t^12). - Zak Seidov, Dec 05 2009

A154635 Ratio of the sum of the bends of the 5-dimensional spheres added in the n-th generation of Apollonian packing to the sum of the bends of the initial configuration of seven mutually tangent spheres.

Original entry on oeis.org

1, 2, 15, 108, 774, 5544, 39708, 284400, 2036952, 14589216, 104492016, 748400832, 5360254560, 38391631488, 274971524544, 1969422407424, 14105550112128, 101027866452480, 723589630947072, 5182549848861696, 37118861005211136, 265855588948518912

Offset: 0

Colin Mallows, Jan 13 2009

			Starting with seven 5-dimensional spheres with bends 0,0,1,1,1,1,1 summing to 5, the first derived generation has seven spheres, with bends 1,1,1,1,1,5/2,5/2 summing to 10. So a(1) = 10/5 = 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).
Index entries for linear recurrences with constant coefficients, signature (8,-6).

Cf. A135849 for dim=2. A137146 for the sum of squares of bends when dim=2. A154636 and A154637 for starting with three spheres in 2 dimensions. A154638-A154645 for results in the three-dimensional case.

CoefficientList[Series[(1 - z) (1 - 5 z)/(1 - 8 z + 6 z^2), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Vec((1-x)*(1-5*x)/(1-8*x+6*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

G.f. (1-x)*(1-5*x) / (1-8*x+6*x^2).
From Colin Barker, Nov 16 2016: (Start)
a(n) = (((4-sqrt(10))^n*(-8+sqrt(10))+(4+sqrt(10))^n*(8+sqrt(10))))/(12*sqrt(10)) for n>0.
a(n) = 8*a(n-1) - 6*a(n-2) for n>2.
(End)

A154641 a(n) is the ratio of the sum of the bends of the spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, using "strategy (a)" to count them (see the reference), to the sum of the bends of the initial five mutually tangent spheres.

Original entry on oeis.org

1, 3, 20, 108, 630, 3570, 20460

Offset: 0

Colin Mallows, Jan 13 2009

In strategy (a) we do not count spheres that can be obtained (by reflection) from the "extra" quintuples of spheres that appear 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.

A154644 a(n) is the ratio of the sum 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 bends of the initial five mutually tangent spheres.

Original entry on oeis.org

1, 3, 20, 174, 1170, 8454

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.

A161410 Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

Original entry on oeis.org

1, 7, 27, 77, 183, 385, 740, 1325, 2242, 3623, 5633, 8474, 12391, 17676, 24670, 33768, 45426, 60164, 78568, 101296, 129083, 162742, 203168, 251346, 308355, 375369, 453663, 544620, 649732, 770602, 908952, 1066628, 1245600, 1447967, 1675965, 1931969, 2218494

Offset: 0

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

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

N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche V.)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A161409, A154638.

Z := Integers();
C := SymmetricMatrix(
[1,
2,1,
3,2,1,
2,3,3,1,
2,2,2,3,1,
2,2,2,2,3,1,
2,3,2,2,2,2,1]);
G := CoxeterGroup(GrpFPCox, C);
f := GrowthFunction(G);
T := PowerSeriesRing(Z, 50);
Eltseq(T!f);
// Corrected by Klaus Brockhaus, Feb 12 2010

CoefficientList[Series[(x^22 + 3 x^21 + 5 x^20 + 7 x^19 + 10 x^18 + 14 x^17 + 17 x^16 + 19 x^15 + 22 x^14 + 25 x^13 + 26 x^12 + 26 x^11 + 26 x^10 + 25 x^9 + 22 x^8 + 19 x^7 + 17 x^6 + 14 x^5 + 10 x^4 + 7 x^3 + 5 x^2 + 3 x + 1) / (x^22 - 4 x^21 + 6 x^20 - 4 x^19 + x^18 - x^15 + 4 x^14 - 6 x^13 + 4 x^12 - 2 x^11 + 4 x^10 - 6 x^9 + 4 x^8 - x^7 + x^4 - 4 x^3 + 6 x^2 - 4 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

G.f.: (x^22 + 3*x^21 + 5*x^20 + 7*x^19 + 10*x^18 + 14*x^17 + 17*x^16 + 19*x^15 + 22*x^14 + 25*x^13 + 26*x^12 + 26*x^11 + 26*x^10 + 25*x^9 + 22*x^8 + 19*x^7 + 17*x^6 + 14*x^5 + 10*x^4 + 7*x^3 + 5*x^2 + 3*x + 1)/(x^22 - 4*x^21 + 6*x^20 - 4*x^19 + x^18 - x^15 + 4*x^14 - 6*x^13 + 4*x^12 - 2*x^11 + 4*x^10 - 6*x^9 + 4*x^8 - x^7 + x^4 - 4*x^3 + 6*x^2 - 4*x + 1)

A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 4, 12, 30, 72, 168, 390, 900, 2076, 4782, 11016, 25368, 58422, 134532, 309804, 713406, 1642824, 3783048, 8711526, 20060676, 46195260, 106377294, 244963080, 564094968, 1298984214, 2991269124, 6888221772, 15862029150, 36526694472, 84112781928, 193692865350

Offset: 0

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

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.
From Bruno Berselli, Dec 28 2015: (Start)
Also, expansion of b(2)*b(3)/(1 - 2*x - 2*x^2 + 3*x^3), where b(k) = (1-x^k)/(1-x).
This is also the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_22 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), page 186.
Index entries for linear recurrences with constant coefficients, signature (2,2,-3).

Cf. similar sequences listed in A265055.

m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(3)/(1-2*x-2*x^2+3*x^3))); // Bruno Berselli, Dec 28 2015 - see Chapovalov et al.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(x^3+2x^2+2x+1)/(3x^3-2x^2-2x+1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
coxG[{3, 3, -2, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^40)); Vec((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (x^3 + 2*x^2 + 2*x + 1)/(3*x^3 - 2*x^2 - 2*x + 1).
From Bruno Berselli, Dec 28 2015: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>3.
a(n) = -2 + ((-7+2*sqrt(13))*(1-sqrt(13))^n + (7+2*sqrt(13))*(1+sqrt(13))^n)/(3*sqrt(13)*2^(n-1)) for n>0. (End)
G.f.: (1+x)*(1-x^3)/(1 -3*x +5*x^3 -3*x^4). - G. C. Greubel, Apr 25 2019

A162783 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 14, 182, 2275, 28392, 353808, 4408950, 54938520, 684572616, 8530235532, 106292493216, 1324476080928, 16503864518232, 205649272719072, 2562528512535264, 31930831990629936, 397879682765894784

Offset: 0

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

nonn

The initial terms coincide with those of A170733, 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..900
Index entries for linear recurrences with constant coefficients, signature (12, 12, -78).

a:=[14, 182, 2275];; for n in [4..20] do a[n]:=12*a[n-1]+12*a[n-2] - 78*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Apr 26 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4), {x,0,20}],x] (* or *) coxG[{3, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(78*t^3 - 12*t^2 - 12*t + 1).

G.f.: (1+x)*(1-x^3)/(1 - 13*x + 90*x^3 - 78*x^4). - G. C. Greubel, Apr 26 2019

A162785 Number of reduced words of length n in Coxeter group on 15 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 15, 210, 2835, 38220, 514605, 6928740, 93285465, 1255955610, 16909618635, 227663487870, 3065158424055, 41267909559240, 555612506386665, 7480515990707760, 100714290692336685, 1355971748798391270

Offset: 0

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

nonn

The initial terms coincide with those of A170734, 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..880
Index entries for linear recurrences with constant coefficients, signature (13, 13, -91).

a:=[15, 210, 2835];; for n in [4..20] do a[n]:=13*a[n-1]+13*a[n-2] -91*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Apr 26 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4), {x, 0, 20}], x] (* or *) coxG[{3, 91, -13}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(91*t^3 - 13*t^2 - 13*t + 1).

G.f.: (1+x)*(1-x^3)/(1 - 14*x + 104*x^3 - 91*x^4). - G. C. Greubel, Apr 26 2019

cp's OEIS Frontend

A164779 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

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A165699 Number of reduced words of length n in Coxeter group on 45 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

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A166377 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

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A154635 Ratio of the sum of the bends of the 5-dimensional spheres added in the n-th generation of Apollonian packing to the sum of the bends of the initial configuration of seven mutually tangent spheres.

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A154641 a(n) is the ratio of the sum of the bends of the spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, using "strategy (a)" to count them (see the reference), to the sum of the bends of the initial five mutually tangent spheres.

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A154644 a(n) is the ratio of the sum 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 bends of the initial five mutually tangent spheres.

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A161410 Number of reduced words of length n in the infinite affine Weyl group (E_6)^{~} on 7 generators.

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Magma

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A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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