A003949 -id:A003949 - cp's OEIS frontend

A163345 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 7, 42, 252, 1512, 9051, 54180, 324345, 1941660, 11623500, 69582660, 416548125, 2493614550, 14927719275, 89362970550, 534960522600, 3202475913000, 19171231408875, 114766238286000, 687034086094125, 4112845750671000

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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, -15).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{5,5,5,5,-15}, {1,7,42,252,1512,9051}, 30] (* G. C. Greubel, Dec 19 2016 *)
coxG[{5,15,-5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 09 2018 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6)) \\ G. C. Greubel, Dec 19 2016

((1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

a(n) = 5*a(n-1)+5*a(n-2)+5*a(n-3)+5*a(n-4)-15*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54411, 326340, 1957305, 11739420, 70410060, 422301600, 2532857460, 15191434125, 91114353750, 546480693675, 3277652052150, 19658522431800, 117906811965600, 707175035973000, 4241455800274875

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,-15).

a:=[7,42,252,1512,9072,54411];; for n in [7..30] do a[n]:=5*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -15*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7) )); // G. C. Greubel, Aug 10 2019

seq(coeff(series((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

coxG[{6,15,-5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 18 2015 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 08 2017 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)) \\ G. C. Greubel, Aug 08 2017

def A163923_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)).list()
A163923_list(30) # G. C. Greubel, Aug 10 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

a(n) = 5*a(n-1)+5*a(n-2)+5*a(n-3)+5*a(n-4)+5*a(n-5)-15*a(n-6). - Wesley Ivan Hurt, Apr 23 2021

A165782 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543851, 423262980, 2539577145, 15237458460, 91424724300, 548548187040, 3291288169680, 19747723302720, 118486305524160, 710917627392000, 4265504529834660

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,5,5,5,5,-15).

a:=[7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543851];; for n in [11..30] do a[n]:=5*Sum([1..9], j-> a[n-j]) -15*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11) )); // G. C. Greubel, Sep 22 2019

seq(coeff(series((1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Sep 22 2019

CoefficientList[Series[(1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 08 2016 *)
coxG[{10, 15, -5}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11)) \\ G. C. Greubel, Aug 07 2017

def A165782_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11) ).list()
A165782_list(30) # G. C. Greubel, Sep 22 2019

G.f.: (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^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).

A166365 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263211, 2539579140, 15237474105, 91424840220, 548549014860, 3291293930400, 19747762629840, 118486570063680, 710919386089920, 4265516110786560

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,5,5,5,5,5,-15).

seq(coeff(series((1+t)*(1-t^11)/(1-6*t+20*t^11-15*t^12), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Mar 13 2020

CoefficientList[Series[(1+t)*(1-t^11)/(1-6*t+20*t^11-15*t^12), {t,0,30}], t] (* G. C. Greubel, May 10 2016 *)
coxG[{11, 15, -5}] (* The coxG program is in A169452 *) (* G. C. Greubel, Mar 13 2020 *)

def A166365_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^11)/(1-6*t+20*t^11-15*t^12) ).list()
A166365_list(30) # G. C. Greubel, Aug 10 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)/(15*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).

A166518 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579371, 15237476100, 91424855865, 548549130780, 3291294758220, 19747768390560, 118486609390800, 710919650629440, 4265517869484480

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,5,5,5,5,5,5,-15).

Cf. A003949, A154638, A169452.

R:=PowerSeriesRing(Integers(), 30);
f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(15,5,x) )); // G. C. Greubel, Aug 03 2024

With[{p=15, q=5}, CoefficientList[Series[(1+t)*(1-t^12)/(1 - (q+1)*t + (p+q)*t^12 - p*t^13), {t,0,40}], t]] (* G. C. Greubel, May 15 2016; Aug 03 2024 *)
coxG[{12, 15, -5, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 03 2024 *)

Vec((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^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)+O(t^99))

def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(15,5,x) ).list()
A166518_list(30) # G. C. Greubel, Aug 03 2024

G.f.: (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^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).
From G. C. Greubel, Aug 03 2024: (Start)
a(n) = 5*Sum_{j=1..11} a(n-j) - 15*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 6*x + 20*x^12 - 15*x^13). (End)

A166878 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476331, 91424857860, 548549146425, 3291294874140, 19747769218380, 118486615151520, 710919689956560, 4265518134024000

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

CoefficientList[Series[(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^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), {t, 0, 50}], t] (* G. C. Greubel, May 27 2016 *)

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)/(15*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).

A167108 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858091, 548549148420, 3291294889785, 19747769334300, 118486615979340, 710919695717280, 4265518173351120

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

CoefficientList[Series[(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^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), {t, 0, 50}], t] (* G. C. Greubel, Jun 03 2016 *)
coxG[{14,15,-5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 08 2018 *)

G.f.: (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^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).

A167652 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^15 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148651, 3291294891780, 19747769349945, 118486616095260, 710919696545100, 4265518179111840

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

CoefficientList[Series[(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^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), {t, 0, 50}], t] (* G. C. Greubel, Jun 18 2016 *)

G.f.: (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^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).

A167898 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892011, 19747769351940, 118486616110905, 710919696661020, 4265518179939660

Offset: 0

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

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

CoefficientList[Series[(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^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), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016 *)

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)/( 15*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).

A168684 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892032, 19747769352171, 118486616112900, 710919696676665, 4265518180055580

Offset: 0

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

nonn

The initial terms coincide with those of A003949, although the two sequences are eventually different.
First disagreement at index 17: a(17) = 19747769352171, A003949(17) = 19747769352192. - Klaus Brockhaus, Mar 30 2011
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 (5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,-15).

Cf. A003949 (g.f.: (1+x)/(1-6*x)).

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) )); // G. C. Greubel, Mar 24 2021

CoefficientList[Series[(1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18), {t, 0, 50}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)
coxG[{17, 15, -5, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Mar 24 2021 *)

def A168684_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) ).list()
A168684_list(40) # G. C. Greubel, Mar 24 2021

G.f.: (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^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+t)*(1-t^17)/(1 - 6*t + 20*t^17 - 15*t^18). - G. C. Greubel, Mar 24 2021

cp's OEIS Frontend

A163345 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

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A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

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A165782 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

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A166365 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

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A166518 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

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A166878 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

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A167108 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

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