A003954 -id:A003954 - cp's OEIS frontend

A163432 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 12, 132, 1452, 15972, 175626, 1931160, 21234840, 233496120, 2567499000, 28231951770, 310435603500, 3413517587700, 37534684133100, 412727480315700, 4538308419052650, 49902767052699000, 548725632894681000

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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..950
Index entries for linear recurrences with constant coefficients, signature (10, 10, 10, 10, -55).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-11*x+65*x^5-55*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-11*x+65*x^5-55*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{10,10,10,10,-55}, {1,12,132,1452,15972, 175626}, 30] (* G. C. Greubel, Dec 23 2016 *)
coxG[{5, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-11*x+65*x^5-55*x^6)) \\ G. C. Greubel, Dec 23 2016

((1+x)*(1-x^5)/(1-11*x+65*x^5-55*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)/(55*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

a(n) = 10*a(n-1)+10*a(n-2)+10*a(n-3)+10*a(n-4)-55*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163957 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932546, 21257280, 233822160, 2571956640, 28290564720, 311185670400, 3422926421970, 37650915208500, 414146038003500, 4555452101075700, 50108275682741100, 551172361422635700

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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..955
Index entries for linear recurrences with constant coefficients, signature (10,10,10,10,10,-55).

a:=[12,132,1452,15972,175692,1932546];; for n in [7..30] do a[n]:=10*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -55*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7) )); // G. C. Greubel, Aug 10 2019

seq(coeff(series((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

CoefficientList[Series[(1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 13 2017 *)
coxG[{6, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 10 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7)) \\ G. C. Greubel, Aug 13 2017

def A163957_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-11*t+65*t^6-55*t^7)).list()
A163957_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)/(55*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

a(n) = -55*a(n-6) + 10*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

A164601 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258666, 233844600, 2572282680, 28295022360, 311244287640, 3423676622520, 37660326891000, 414262320281370, 4556871492422700, 50125432079728500, 551378055176107500

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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..950
Index entries for linear recurrences with constant coefficients, signature (10,10,10,10,10,10,-55).

a:=[12, 132, 1452, 15972, 175692, 1932612, 21258666];; for n in [8..30] do a[n]:=10*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -55*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-11*t+65*t^7-55*t^8) )); // G. C. Greubel, Aug 28 2019

seq(coeff(series((1+t)*(1-t^7)/(1-11*t+65*t^7-55*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

CoefficientList[Series[(1+t)*(1-t^7)/(1-11*t+65*t^7-55*t^8), {t, 0, 30}], t] (* G. C. Greubel, Aug 11 2017 *)
coxG[{7, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-11*t+65*t^7-55*t^8)) \\ G. C. Greubel, Aug 11 2017

def A164601_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-11*t+65*t^7-55*t^8)).list()
A164601_list(30) # G. C. Greubel, Aug 28 2019

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(55*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

a(n) = -55*a(n-7) + 10*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

A165266 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306506, 28295370840, 311249071320, 3423739697400, 37661135713080, 414272482302360, 4556997189369240, 50126967807537720, 551396631852151800

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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..955
Index entries for linear recurrences with constant coefficients, signature (10,10,10,10,10,10,10,10,-55).

a:=[12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306506];; for n in [10..30] do a[n]:=10*Sum([1..8], j-> a[n-j]) -55*a[n-9]; od; Concatenation([1], a); # G. C. Greubel, Sep 25 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10) )); // G. C. Greubel, Sep 25 2019

seq(coeff(series((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 25 2019

CoefficientList[Series[(1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10), {t,0,30}], t] (* or *) coxG[{9, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 25 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10)) \\ G. C. Greubel, Sep 25 2019

def A165266_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-11*t+65*t^9-55*t^10)).list()
A165266_list(30) # G. C. Greubel, Sep 25 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)/(55*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

A165807 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372226, 311249093760, 3423740023440, 37661140170720, 414272540919600, 4556997939574080, 50126977219358160, 551396748137415840

Offset: 0

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

nonn

The initial terms coincide with those of A003954, although the two sequences are eventually different.
First disagreement at index 10: a(10) = 28295372226, A003954(10) = 28295372292. - Klaus Brockhaus, Jun 14 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 (10,10,10,10,10,10,10,10,10,-55).

Cf. A003954 (G.f.: (1+x)/(1-11*x)).

a:=[12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372226];; for n in [11..20] do a[n]:=10*Sum([1..9], j-> a[n-j]) -55*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 23 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11) )); // G. C. Greubel, Sep 23 2019

seq(coeff(series((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 23 2019

With[{num=Total[2t^Range[9]]+1+t^10,den=Total[-10 t^Range[9]]+1+ 55t^10}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Jun 14 2011 *)
CoefficientList[Series[(1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), {t, 0, 30}], t] (* or *) coxG[{10, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 23 2019 *)

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)) \\ G. C. Greubel, Sep 23 2019

def A165807_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)).list()
A165807_list(20) # G. C. Greubel, Sep 23 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)/(55*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

A166372 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095146, 3423740045880, 37661140496760, 414272545377240, 4556997998191320, 50126977969563000, 551396757549236280

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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 (10,10,10,10,10,10,10,10,10,10,-55).

Cf. A003954, A154638, A169452.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^11)/(1-11*x+65*x^11-55*x^12) )); // G. C. Greubel, Dec 06 2024

CoefficientList[Series[(1+t)*(1-t^11)/(1-11*t+65*t^11-55*t^12), {t,0,50}], t] (* G. C. Greubel, May 10 2016; Dec 06 2024 *)
coxG[{11,55,-10,40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 06 2024 *)

def A166372_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-11*x+65*x^11-55*x^12) ).list()
print(A166372_list(40)) # G. C. Greubel, Dec 06 2024

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)/(55*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).
From G. C. Greubel, Dec 06 2024: (Start)
a(n) = 10*Sum_{j=1..10} a(n-j) - 55*a(n-11).
G.f.: (1+x)*(1-x^11)/(1 - 11*x + 65*x^11 - 55*x^12). (End)

A166557 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047266, 37661140519200, 414272545703280, 4556998002648960, 50126978028180240, 551396758299441120, 6065364341177895600, 66719007751681327680

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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 (10,10,10,10,10,10,10,10,10,10,10,-55).

Cf. A003954, A154638, A169452.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^12)/(1-11*x+65*x^12-55*x^13) )); // G. C. Greubel, Dec 03 2024

CoefficientList[Series[(1+t)*(1-t^12)/(1-11*t+65*t^12-55*t^13), {t,0,50}], t]
(* G. C. Greubel, May 17 2016; Dec 03 2024 *)
coxG[{12,55,-10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 03 2024 *)

def A166557_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-11*x+65*x^12-55*x^13) ).list()
A166557_list(40) # G. C. Greubel, Dec 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)/(55*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t +1).
From G. C. Greubel, Dec 03 2024: (Start)
a(n) = 10*Sum_{j=1..11} a(n-j) - 55*a(n-12).
G.f.: (1+t)*(1 - t^12)/(1 - 11*t + 65*t^12 - 55*t^13). (End)

A166951 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520586, 414272545725720, 4556998002975000, 50126978032637880, 551396758358058360

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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 (10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, -55).

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)/(55*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1), {t, 0, 50}], t] (* G. C. Greubel, May 29 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)/(55*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

A167113 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727106, 4556998002997440, 50126978032963920, 551396758362516000

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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 (10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, -55).

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)/ (55*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 03 2016 *)

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)/(55*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

A167665 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^15 = I.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727172, 4556998002998826, 50126978032986360, 551396758362842040

Offset: 0

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

nonn

The initial terms coincide with those of A003954, 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 (10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, -55).

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)/(55*t^15 - 10*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 19 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)/(55*t^15 - 10*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).

cp's OEIS Frontend

A163432 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

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A163957 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

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A164601 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

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A165266 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

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A165807 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

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GAP

Magma

Maple

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PARI

Sage

Formula

A166372 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

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Magma

Mathematica