cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A163231 Number of reduced words of length n in Coxeter group on 45 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 45, 1980, 87120, 3832290, 168577200, 7415481150, 326196882000, 14348955088710, 631190926398780, 27765226324720170, 1221354364616557380, 53725709508796162530, 2363320544672336677560, 103959241263364038810390
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^4)/(1-44*x+989*x^4-946*x^5) )); // G. C. Greubel, Apr 30 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(946*t^4-43*t^3-43*t^2 - 43*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[ {43, 43, 43, -946}, {45,1980,87120,3832290}, 20]] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 946, -43}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 30 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(946*t^4-43*t^3 - 43*t^2-43*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-44*x+989*x^4-946*x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(946*t^4 - 43*t^3 - 43*t^2 - 43*t + 1).
a(n) = 43*a(n-1)+43*a(n-2)+43*a(n-3)-946*a(n-4). - Wesley Ivan Hurt, May 06 2021

A163232 Number of reduced words of length n in Coxeter group on 46 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 46, 2070, 93150, 4190715, 188535600, 8482007160, 381596054400, 17167581467190, 772350369021000, 34747182860785560, 1563237055602189000, 70328294002955286540, 3163991615757072698400, 142344458748855549948960
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170765, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[46,2070,93150,4190715];; for n in [5..20] do a[n]:=44*(a[n-1] +a[n-2] +a[n-3]) -990*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, May 01 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^4)/(1-45*x+1034*x^4-990*x^5) )); // G. C. Greubel, May 01 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(990*t^4-44*t^3-44*t^2 - 44*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[ {44, 44, 44, -990}, {46,2070,93150,4190715}, 20]] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 990, -44}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 01 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(990*t^4-44*t^3 - 44*t^2-44*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-45*x+1034*x^4-990*x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(990*t^4 - 44*t^3 - 44*t^2 - 44*t + 1).
a(n) = 44*a(n-1)+44*a(n-2)+44*a(n-3)-990*a(n-4). - Wesley Ivan Hurt, May 10 2021

A163265 Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 47, 2162, 99452, 4573711, 210340980, 9673398765, 444871172700, 20459237269140, 940902479912925, 43271284508242650, 1990008638480367675, 91518761835509986350, 4208868045065726973000, 193562170919821248573375
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170766, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[47,2162,99452,4573711];; for n in [5..20] do a[n]:=45*(a[n-1]+a[n-2] +a[n-3]-23*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, May 01 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-46*x+1080*x^4-1035*x^5) )); // G. C. Greubel, May 01 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(1035*t^4-45*t^3-45*t^2 - 45*t+1), {t,0,20}], t] (* or *) LinearRecurrence[ {45, 45, 45, -1035}, {1,47,2162,99452,4573711}, 20] (* G. C. Greubel, Dec 12 2016 *)
coxG[{4, 1035, -45}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 01 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(1035*t^4-45*t^3- 45*t^2-45*t+1)) \\ G. C. Greubel, Dec 12 2016
  • Sage
    ((1+x)*(1-x^4)/(1-46*x+1080*x^4-1035*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1035*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).
a(n) = 45*a(n-1)+45*a(n-2)+45*a(n-3)-1035*a(n-4). - Wesley Ivan Hurt, May 10 2021

A163266 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 48, 2256, 106032, 4982376, 234118656, 11001086208, 516933992448, 24290397127896, 1141390199234256, 53633194222120752, 2520189436004377296, 118422087020288430408, 5564578001118314478240, 261475955285477822620512, 12286587622406034842484384, 577338880885792093267553208
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170767, although the two sequences are eventually different.
Computed with Magma using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[48,2256,106032,4982376];; for n in [5..20] do a[n]:=46*(a[n-1] +a[n-2] +a[n-3]) -1081*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, May 01 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-47*x+1127*x^4-1081*x^5) )); // G. C. Greubel, May 01 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(1081*t^4-46*t^3-46*t^2 - 46*t+1), {t,0,20}], t] (* or *) LinearRecurrence[ {46,46,46,-1081}, {1,48,2256,106032,4982376}, 20] (* G. C. Greubel, Dec 12 2016 *)
coxG[{4, 1081, -46}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 01 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(1081*t^4-46*t^3 - 46*t^2-46*t+1)) \\ G. C. Greubel, Dec 12 2016
  • Sage
    ((1+x)*(1-x^4)/(1-47*x+1127*x^4-1081*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1081*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).
a(n) = 46*a(n-1)+46*a(n-2)+46*a(n-3)-1081*a(n-4). - Wesley Ivan Hurt, May 10 2021

A163287 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 49, 2352, 112896, 5417832, 259999488, 12477267096, 598778820864, 28735144795560, 1378987562102976, 66177035471527512, 3175808211876089664, 152405705797427455464, 7313885981134376257152, 350990324575741067673624
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170768, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[49,2352,112896,5417832];; for n in [5..20] do a[n]:=47*(a[n-1]+a[n-2] +a[n-3] -24*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, May 01 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-48*x+1175*x^4-1128*x^5) )); // G. C. Greubel, May 01 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(1128*t^4-47*t^3-47*t^2 - 47*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{47, 47, 47, -1128}, {1,49,2352,112896,5417832}, 20] (* G. C. Greubel, Dec 17 2016 *)
coxG[{4, 1128, -47}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 01 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(1128*t^4-47*t^3 - 47*t^2-47*t+1)) \\ G. C. Greubel, Dec 17 2016
  • Sage
    ((1+x)*(1-x^4)/(1-48*x+1175*x^4-1128*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1128*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).
a(n) = 47*a(n-1)+47*a(n-2)+47*a(n-3)-1128*a(n-4). - Wesley Ivan Hurt, May 10 2021

A163314 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 3, 6, 12, 24, 45, 84, 159, 300, 564, 1062, 2001, 3768, 7095, 13362, 25164, 47388, 89241, 168060, 316491, 596016, 1122420, 2113746, 3980613, 7496304, 14117067, 26585310, 50065548, 94283616, 177555237, 334372644, 629691735, 1185837684
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • GAP
    a:=[3,6,12,24];; for n in [5..40] do a[n]:=2*a[n-1]-a[n-2]+ 2*a[n-3]-a[n-4]; od; Concatenation([1], a); # G. C. Greubel, May 12 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1), {t,0,40} ], t] (* or *) LinearRecurrence[{2,-1,2,-1}, {1,3,6,12,24}, 40] (* G. C. Greubel, Dec 18 2016 *)
  • PARI
    my(t='t+O('t^40)); Vec((t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1)) \\ G. C. Greubel, Dec 18 2016
  • Sage
    ((x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^4 + t^3 + t^2 + t + 1)/(t^4 - 2*t^3 + t^2 - 2*t + 1).
a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-a(n-4). - Wesley Ivan Hurt, May 10 2021

A163315 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 4, 12, 36, 108, 318, 936, 2760, 8136, 23976, 70662, 208260, 613788, 1808964, 5331420, 15712878, 46309320, 136483800, 402247944, 1185513624, 3493970742, 10297504260, 30349021740, 89445276900, 263615006412, 776931706398
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6), {x,0,30}], x] (* or *) Join[{1}, LinearRecurrence[{2,2,2,2,-3}, {1,4,12,36,108,318}, 30]] (* G. C. Greubel, Dec 18 2016 *)
coxG[{4, 3, -2}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6)) \\ G. C. Greubel, Dec 18 2016
  • Sage
    ((1+x)*(1-x^5)/(1-3*x+5*x^5-3*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).
a(n) = 2*a(n-1)+2*a(n-2)+2*a(n-3)+2*a(n-4)-3*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163316 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1270, 5040, 20010, 79440, 315360, 1251930, 4969980, 19730070, 78325380, 310939920, 1234384470, 4900319640, 19453527810, 77227563240, 306581745960, 1217083163130, 4831636082580, 19180864497870, 76145131089180
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A003947, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{3,3,3,3,-6}, {1,5,20,80,320,1270}, 30] (* G. C. Greubel, Dec 18 2016 *)
coxG[{5, 6, -3}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)) \\ G. C. Greubel, Dec 18 2016
  • Sage
    ((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(6*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).
a(n) = 3*a(n-1)+3*a(n-2)+3*a(n-3)+3*a(n-4)-6*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163317 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3735, 18600, 92640, 461400, 2298000, 11445210, 57003000, 283904040, 1413987000, 7042377000, 35074632060, 174689570400, 870043225440, 4333259349600, 21581843340000, 107488595621160, 535348070440800
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A003948, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{4,4,4,4,-10}, {1,6,30,150,750,3735}, 30] (* G. C. Greubel, Dec 18 2016 *)
coxG[{5, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6)) \\ G. C. Greubel, Dec 18 2016
  • Sage
    ((1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).
a(n) = 4*a(n-1)+4*a(n-2)+4*a(n-3)+4*a(n-4)-10*a(n-5). - Wesley Ivan Hurt, May 10 2021

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.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9051, 54180, 324345, 1941660, 11623500, 69582660, 416548125, 2493614550, 14927719275, 89362970550, 534960522600, 3202475913000, 19171231408875, 114766238286000, 687034086094125, 4112845750671000
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    ((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

Formula

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
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