A154638 -id:A154638 - cp's OEIS frontend

A167049 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516992, 67838877305856, 1221099791505408, 21979796247097173, 395636332447746036, 7121453984059373415

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, -153).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14), {x, 0, 20}], x] (* G. C. Greubel, May 31 2016, modified Apr 26 2019 *)
coxG[{13, 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^13)/(1-18*x+170*x^13-153*x^14)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (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^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^13)/(1 - 18*x + 170*x^13 - 153*x^14). - G. C. Greubel, Apr 26 2019
a(n) = -153*a(n-13) + 17*Sum_{k=1..12} a(n-k). - Wesley Ivan Hurt, May 06 2021

A167942 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743552, 146596599332352, 3811511582641152, 99099301148669952, 2576581829865418752, 66991127576500887552, 1741769316989023076352

Offset: 0

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

nonn

The initial terms coincide with those of A170746, 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 (25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,-325).

Cf. A154638, A169452, A170758.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-26*x+350*x^16-325*x^17) )); // G. C. Greubel, Sep 08 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-26*t+350*t^16-325*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 08 2023 *)
coxG[{16,325,-25}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 28 2018 *)

def A167942_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-26*x+350*x^16-325*x^17) ).list()
A167942_list(40) # G. C. Greubel, Sep 08 2023

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)/( 325*t^16 - 25*t^15 - 25*t^14 - 25*t^13 - 25*t^12 - 25*t^11 - 25*t^10 - 25*t^9 - 25*t^8 - 25*t^7 - 25*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).
From G. C. Greubel, Sep 08 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 26*t + 350*t^16 - 325*t^17).
a(n) = 25*Sum_{j=1..15} a(n-j) - 325*a(n-16). (End)

A154639 a(n) is the number of reduced words of length n (i.e., all possible length-reducing cancellations have been applied) in the generators of the "Apollonian reflection group" in three dimensions. This is a Coxeter group with five generators, satisfying the identities (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 5, 20, 80, 300, 1140, 4260

Offset: 0

Colin Mallows, Jan 13 2009

ABA and BAB are equal, but are counted as distinct reduced words.

			All 80 squarefree words of length 3 are counted, so a(3) = 80.

R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. Yan, Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions., Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.
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.

A154640 a(n) is the number of spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, starting with five mutually tangent spheres and using "strategy (a)" to count them (see the reference).

Original entry on oeis.org

5, 5, 20, 60, 210, 690, 3330

Offset: 0

Colin Mallows, Jan 13 2009

In strategy (a) we do not count spheres that are generated (by reflection) from quintuples that were not generated in the previous generation.

			For a(3), we apply reflection only to the 20 quintuples that were generated in the second generation, ignoring the 10 "extra" quintuples (which will appear as ABA = BAB in the third generation).

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.

A162760 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 11, 110, 1045, 9900, 93555, 884070, 8353125, 78924780, 745717995, 7045894350, 66572896005, 629011803420, 5943197049075, 56154099352230, 530570136457845, 5013074255082300, 47365865053010955, 447534797632236270

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, 9, -45).

I:=[1,11,110,1045]; [n le 4 select I[n] else 9*Self(n-1) +9*Self(n-2)-45*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 01 2017

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4) )); // G. C. Greubel, Apr 26 2019

Join[{1}, LinearRecurrence[{9, 9, -45}, {11, 110, 1045}, 19]] (* Vincenzo Librandi, Apr 01 2017 *)
CoefficientList[Series[(1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4), {x,0,20}],x] (* or *) coxG[{3, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^3)/(1-10*x+54*x^3-45*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)/(45*t^3 - 9*t^2 - 9*t + 1).

G.f.: (1+x)*(1-x^3)/(1 - 10*x + 54*x^3 - 45*x^4). - G. C. Greubel, Apr 26 2019

A162851 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 37, 1332, 47286, 1678320, 59557050, 2113447770, 74997827100, 2661373678950, 94441530616650, 3351353019273000, 118926143828399250, 4220214225380039250, 149758560520153357500, 5314333645481777358750, 188584492248078150341250

Offset: 0

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

nonn

The initial terms coincide with those of A170756, 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..643
Index entries for linear recurrences with constant coefficients, signature (35, 35, -630).

a:=[37, 1332, 47286];; for n in [4..20] do a[n]:=35*a[n-1]+ 35*a[n-2]-630*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Apr 26 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 +2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1))); // G. C. Greubel, Oct 24 2018

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1), {t, 0, 20}], t] (* or *) LinearRecurrence[{35, 35, -630}, {1, 37, 1332}, 20] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 630, -35}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1)) \\ G. C. Greubel, Oct 24 2018

((1+x)*(1-x^3)/(1-36*x+665*x^3-630*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)/(630*t^3 - 35*t^2 - 35*t + 1).
G.f.: (1+x)*(1-x^3)/(1 - 36*x + 665*x^3 - 630*x^4). - G. C. Greubel, Apr 26 2019
a(n) = 35*a(n-1)+35*a(n-2)-630*a(n-3). - Wesley Ivan Hurt, May 05 2021

A162858 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 38, 1406, 51319, 1872792, 68331600, 2493179658, 90967125816, 3319062151464, 121100596329852, 4418523599533920, 161215975658220768, 5882188976123487336, 214619841546851901024, 7830703259038738949472

Offset: 0

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

nonn

The initial terms coincide with those of A170757, 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..638
Index entries for linear recurrences with constant coefficients, signature (36, 36, -666).

a:=[38,1406,51319];; for n in [4..20] do a[n]:=36*a[n-1]+36*a[n-2]-666*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 25 2018

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1))); // G. C. Greubel, Oct 24 2018

seq(coeff(series((x^3+2*x^2+2*x+1)/(666*x^3-36*x^2-36*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 666, -36}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2019 *)

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1)) \\ G. C. Greubel, Oct 24 2018

((1+x)*(1-x^3)/(1 -37*x +702*x^3 -666*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)/(666*t^3 - 36*t^2 - 36*t + 1).
a(n) = 36*a(n-1) + 36*a(n-2) - 666*a(n-3), n > 0. - Muniru A Asiru, Oct 25 2018
G.f.: (1+x)*(1-x^3)/(1 - 37*x + 702*x^3 - 666*x^4). - G. C. Greubel, Apr 27 2019

A162871 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 39, 1482, 55575, 2083692, 78111033, 2928135600, 109766289945, 4114781688966, 154249795892907, 5782323668697966, 216760526662519203, 8125647855742321632, 304604136609884440797, 11418619374984439210164

Offset: 0

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

nonn

The initial terms coincide with those of A170758, 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..615
Index entries for linear recurrences with constant coefficients, signature (37, 37, -703).

a:=[39,1482,55575];; for n in [4..15] do a[n]:=37*a[n-1]+37*a[n-2]-703*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1))); // G. C. Greubel, Oct 24 2018

seq(coeff(series((x^3+2*x^2+2*x+1)/(703*x^3-37*x^2-37*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

coxG[{3,703,-37}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 25 2018 *)
CoefficientList[Series[(t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(703*t^3-37*t^2-37*t+1)) \\ G. C. Greubel, Oct 24 2018

((1+x)*(1-x^3)/(1 -38*x +740*x^3 -703*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 27 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(703*t^3 - 37*t^2 - 37*t + 1).
a(n) = 37*a(n-1) + 37*a(n-2) - 703*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018
G.f.: (1+x)*(1-x^3)/(1 - 38*x + 740*x^3 - 703*x^4). - G. C. Greubel, Apr 27 2019

A162877 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 40, 1560, 60060, 2311920, 88979280, 3424561140, 131801403240, 5072652999960, 195231667516860, 7513899339838320, 289188142406526480, 11130010920731869140, 428361764988438838440, 16486399071025250766360

Offset: 0

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

nonn

The initial terms coincide with those of A170759, 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..615
Index entries for linear recurrences with constant coefficients, signature (38, 38, -741).

a:=[40,1560,60060];; for n in [4..20] do a[n]:=38*a[n-1]+38*a[n-2] -741*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(741*t^3-38*t^2-38*t+1))); // G. C. Greubel, Oct 24 2018

seq(coeff(series((x^3+2*x^2+2*x+1)/(741*x^3-38*x^2-38*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

coxG[{3,741,-38}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 29 2017 *)
CoefficientList[Series[(t^3+2*t^2+2*t+1)/(741*t^3-38*t^2-38*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(741*t^3-38*t^2-38*t+1)) \\ G. C. Greubel, Oct 24 2018

((1+x)*(1-x^3)/(1 -39*x +779*x^3 -741*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 27 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(741*t^3 - 38*t^2 - 38*t + 1).
a(n) = 38*a(n-1) + 38*a(n-2) - 741*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018
G.f.: (1+x)*(1-x^3)/(1 - 39*x + 779*x^3 - 741*x^4). - G. C. Greubel, Apr 27 2019

A162878 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 41, 1640, 64780, 2558400, 101024820, 3989217180, 157523886000, 6220211664420, 245620097065980, 9698903409405600, 382984651654144020, 15123074971766970780, 597171180654087109200, 23580747941118076783620

Offset: 0

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

nonn

The initial terms coincide with those of A170760, 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..615
Index entries for linear recurrences with constant coefficients, signature (39, 39, -780).

a:=[41,1640,64780];; for n in [4..20] do a[n]:=39*a[n-1]+39*a[n-2] -780*a[n-3]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 + 2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1))); // G. C. Greubel, Oct 24 2018

seq(coeff(series((x^3+2*x^2+2*x+1)/(780*x^3-39*x^2-39*x+1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 24 2018

CoefficientList[Series[(t^3+2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1), {t, 0, 20}], t] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 780, -39}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2019 *)

my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(780*t^3-39*t^2-39*t+1)) \\ G. C. Greubel, Oct 24 2018

((1+x)*(1-x^3)/(1-40*x+819*x^3-780*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 27 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(780*t^3 - 39*t^2 - 39*t + 1).
a(n) = 39*a(n-1) + 39*a(n-2) - 780*a(n-3), n > 0. - Muniru A Asiru, Oct 24 2018
G.f.: (1+x)*(1-x^3)/(1 - 40*x + 819*x^3 - 780*x^4). - G. C. Greubel, Apr 27 2019

cp's OEIS Frontend

A167049 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

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A167942 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A154639 a(n) is the number of reduced words of length n (i.e., all possible length-reducing cancellations have been applied) in the generators of the "Apollonian reflection group" in three dimensions. This is a Coxeter group with five generators, satisfying the identities (S_i)^2 = (S_i S_j)^3 = I.

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A154640 a(n) is the number of spheres that are added in the n-th generation of Apollonian packing of three-dimensional spheres, starting with five mutually tangent spheres and using "strategy (a)" to count them (see the reference).

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A162760 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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A162851 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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A162858 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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