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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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

A163347 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19180, 134064, 937104, 6550320, 45786384, 320044452, 2237094216, 15637173048, 109303031880, 764022547512, 5340478146444, 37329666414768, 260932440209616, 1823904280240560, 12748996716570576
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A003950, 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)/(1-7*x+27*x^5-21*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,6,6,6,-21}, {1,8,56,392,2744,19180}, 30] (* G. C. Greubel, Dec 19 2016 *)
coxG[{5, 21, -6}] (* 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)/(1-7*x+27*x^5-21*x^6)) \\ G. C. Greubel, Dec 19 2016
  • Sage
    ((1+x)*(1-x^5)/(1-7*x+27*x^5-21*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)/(21*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
a(n) = 6*a(n-1)+6*a(n-2)+6*a(n-3)+6*a(n-4)-21*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163391 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 9, 72, 576, 4608, 36828, 294336, 2352420, 18801216, 150264576, 1200956652, 9598382640, 76712967828, 613111567824, 4900159716480, 39163451657148, 313005296651040, 2501626174048260, 19993698450611424, 159795249138713664
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A003951, 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)/(1-8*x+35*x^5-28*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-8*x+35*x^5-28*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{7,7,7,7,-28}, {1,9,72,576,4608,36828}, 30] (* G. C. Greubel, Dec 21 2016 *)
coxG[{5, 28, -7}] (* 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)/(1-8*x+35*x^5-28*x^6)) \\ G. C. Greubel, Dec 21 2016
  • Sage
    ((1+x)*(1-x^5)/(1-8*x+35*x^5-28*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)/(28*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).
a(n) = 7*a(n-1)+7*a(n-2)+7*a(n-3)+7*a(n-4)-28*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163397 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 10, 90, 810, 7290, 65565, 589680, 5303520, 47699280, 429001920, 3858394860, 34701968160, 312105587040, 2807042441760, 25246223065440, 227061682284240, 2042167156174080, 18367021030590720, 165190915209012480
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-9*x+44*x^5-36*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-9*x+44*x^5-36*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{8,8,8,8,-36}, {1,10,90,810,7290,65565}, 30] (* G. C. Greubel, Dec 21 2016 *)
coxG[{5, 36, -8}] (* 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)/(1-9*x+44*x^5-36*x^6)) \\ G. C. Greubel, Dec 21 2016
  • Sage
    ((1+x)*(1-x^5)/(1-9*x+44*x^5-36*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)/(36*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).
a(n) = 8*a(n-1)+8*a(n-2)+8*a(n-3)+8*a(n-4)-36*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163404 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 109945, 1098900, 10983555, 109781100, 1097266500, 10967222970, 109617836625, 1095634704780, 10950913128375, 109454819042250, 1094005337374620, 10934627535602100, 109292043884611005
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1 + x)*(1-x^5)/(1-10*x+54*x^5-45*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{9, 9, 9, 9, -45}, {1, 11, 110, 1100, 11000, 109945}, 30] (* G. C. Greubel, Dec 21 2016 *)
coxG[{5, 45, -9}] (* 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)/(1-10*x+54*x^5-45*x^6)) \\ G. C. Greubel, Dec 21 2016
  • Sage
    ((1+x)*(1-x^5)/(1-10*x+54*x^5-45*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)/(45*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
a(n) = 9*a(n-1)+9*a(n-2)+9*a(n-3)+9*a(n-4)-45*a(n-5). - Wesley Ivan Hurt, May 10 2021

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.

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175626, 1931160, 21234840, 233496120, 2567499000, 28231951770, 310435603500, 3413517587700, 37534684133100, 412727480315700, 4538308419052650, 49902767052699000, 548725632894681000
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

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

Formula

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

A163438 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 13, 156, 1872, 22464, 269490, 3232944, 38784174, 465276240, 5581708704, 66961236342, 803303685756, 9636871221978, 115609188148740, 1386911174446512, 16638146470934274, 199600322709006648
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-12*x+77*x^5-66*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-12*x+77*x^5-66*x^6), {x, 0, 10}], x] (* or *) LinearRecurrence[{11, 11, 11, 11, -66}, {1, 13, 156, 1872, 22464, 269490}, 30] (* G. C. Greubel, Dec 23 2016 *)
coxG[{5, 66, -11}] (* 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)/(1-12*x+77*x^5-66*x^6)) \\ G. C. Greubel, Dec 23 2016
  • Sage
    ((1+x)*(1-x^5)/(1-12*x+77*x^5-66*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)/(66*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).
a(n) = 11*a(n-1)+11*a(n-2)+11*a(n-3)+11*a(n-4)-66*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163439 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399763, 5195736, 67529280, 877681896, 11407280976, 148261073142, 1926957516120, 25044775341768, 325508355356184, 4230650423530440, 54986001777229068, 714656161291232160
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6) )); // G. C. Greubel, May 12 2019
  • Mathematica
    CoefficientList[Series[(1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{12, 12, 12, 12, -78}, {1, 14, 182, 2366, 30758, 399763}, 30] (* G. C. Greubel, Dec 23 2016 *)
coxG[{5, 78, -12}] (* 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)/(1-13*x+90*x^5-78*x^6)) \\ G. C. Greubel, Dec 23 2016
  • Sage
    ((1+x)*(1-x^5)/(1-13*x+90*x^5-78*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)/(78*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
a(n) = 12*a(n-1)+12*a(n-2)+12*a(n-3)+12*a(n-4)-78*a(n-5). - Wesley Ivan Hurt, May 10 2021
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