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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A163207 Number of reduced words of length n in Coxeter group on 29 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 29, 812, 22736, 636202, 17802288, 498146166, 13939191504, 390048294510, 10914382803996, 305407698579522, 8545958486918244, 239134137088822794, 6691482951706744632, 187241958166564053774, 5239429159586654676168
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170748, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[29,812,22736,636202];; for n in [5..20] do a[n]:=27*(a[n-1] +a[n-2]+a[n-3] -14*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-28*x+405*x^4-378*x^5) )); // G. C. Greubel, Apr 28 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(378*t^4-27*t^3-27*t^2 - 27*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{27,27,27,-378}, {1,29, 812,22736,636202}, 20] (* G. C. Greubel, Dec 10 2016 *)
coxG[{4, 378, -27}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-28*x+405*x^4-378*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019
  • Sage
    ((1+x)*(1-x^4)/(1-28*x+405*x^4-378*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(378*t^4 - 27*t^3 - 27*t^2 - 27*t + 1).
From G. C. Greubel, Apr 28 2019: (Start)
a(n) = 27*(a(n-1) + a(n-2) + a(n-3) -14*a(n-4)).
G.f.: (1+x)*(1-x^4)/(1 - 28*x + 405*x^4 - 378*x^5). (End)

A163208 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731235, 21193200, 614237400, 17802288000, 515959239390, 14953916974920, 433405617680280, 12561286100120520, 364060598322527820, 10551476830837383840, 305810801346502707360, 8863237603561904401440
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[30,870,25230,731235];; for n in [5..20] do a[n]:=28*(a[n-1] + a[n-2]+a[n-3]) -406*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5) )); // G. C. Greubel, Apr 28 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(406*t^4-28*t^3-28*t^2- 28*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{28,28,28,-406}, {1,30, 870,25230,731235}, 20] (* G. C. Greubel, Dec 10 2016 *)
coxG[{4, 406, -28}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019
  • Sage
    ((1+x)*(1-x^4)/(1-29*x+434*x^4-406*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(406*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).
From G. C. Greubel, Apr 28 2019: (Start)
a(n) = 28*(a(n-1) + a(n-2) + a(n-3)) - 406*a(n-4).
G.f.: (1+x)*(1-x^4)/(1 - 29*x + 434*x^4 - 406*x^5). (End)

A163214 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 31, 930, 27900, 836535, 25082100, 752044965, 22548807900, 676088221260, 20271372436125, 607803134933490, 18223958540698875, 546414860017738110, 16383333982098029400, 491226816855341457015, 14728612983261055500600
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170750, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[31,930,27900,836535];; for n in [5..20] do a[n]:=29*(a[n-1]+ a[n-2] +a[n-3] -15*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5) )); // G. C. Greubel, Apr 28 2019
  • Mathematica
    coxG[{4,435,-29}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 24 2016 *)
CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(435*t^4-29*t^3-29*t^2 - 29*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{29,29,29,-435}, {1,31, 930,27900,836535}, 20] (* G. C. Greubel, Dec 10 2016 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)) \\ G. C. Greubel, Dec 10 2016, modified Apr 28 2019
  • Sage
    ((1+x)*(1-x^4)/(1-30*x+464*x^4-435*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(435*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).
a(n) = 29*(a(n-1) + a(n-2) + a(n-3) - 15*a(n-4)). - G. C. Greubel, Apr 28 2019

A163216 Number of reduced words of length n in Coxeter group on 33 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 33, 1056, 33792, 1080816, 34569216, 1105674768, 35364307968, 1131105025776, 36177678932736, 1157120181575952, 37009757234816256, 1183733679862288368, 37860973146888460800, 1210959282493490855952, 38731766829339020895744
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170752, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • GAP
    a:=[33,1056,33792,1080816];; for n in [5..20] do a[n]:=31*(a[n-1]+ a[n-2]+a[n-3]-16*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-32*x+527*x^4-496*x^5) )); // G. C. Greubel, Apr 28 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(496*t^4-31*t^3-31*t^2 - 31*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{31,31,31,-496}, {1,33, 1056,33792,1080816}, 20] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 496, -31}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-32*x+527*x^4-496*x^5)) \\ G. C. Greubel, Dec 11 2016, modified Apr 28 2019
  • Sage
    ((1+x)*(1-x^4)/(1-32*x+527*x^4-496*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(496*t^4 - 31*t^3 - 31*t^2 - 31*t + 1).
From G. C. Greubel, Apr 28 2019: (Start)
a(n) = 31*(a(n-1) + a(n-2) + a(n-3) - 16*a(n-4)).
G.f.: (1+x)*(1-x^4)/(1 - 32*x + 527*x^4 - 496*x^5). (End)

A163217 Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 34, 1122, 37026, 1221297, 40284288, 1328771136, 43829305344, 1445702699760, 47686274735616, 1572924224543232, 51882656590093824, 1711341215834452224, 56448319139710451712, 1861938872397761101824, 61415759005426222645248
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170753, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A154638, A170753.

Programs

  • GAP
    a:=[34,1122,37026,1221297];; for n in [5..20] do a[n]:=32*(a[n-1]+ a[n-2]+a[n-3]) -528*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5) )); // G. C. Greubel, Apr 28 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(528*t^4-32*t^3-32*t^2 - 32*t+1), {t,0,20}], t] (* or *)
LinearRecurrence[{32, 32, 32, -528}, {1, 34, 1122, 37026, 1221297}, 20] (* G. C. Greubel, Dec 11 2016; simplified by Georg Fischer, Apr 08 2019 *)
coxG[{4,528,-32}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 06 2018 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5)) \\ G. C. Greubel, Dec 11 2016, modified Apr 28 2019
  • Sage
    ((1+x)*(1-x^4)/(1-33*x+560*x^4-528*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(528*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).
From G. C. Greubel, Apr 28 2019: (Start)
a(n) = 32*(a(n-1) + a(n-2) + a(n-3)) - 528*a(n-4).
G.f.: (1+x)*(1-x^4)/(1 - 33*x + 560*x^4 - 528*x^5). (End)

A163218 Number of reduced words of length n in Coxeter group on 35 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 35, 1190, 40460, 1375045, 46731300, 1588176975, 53974651500, 1834344072330, 62340711467265, 2118667029023160, 72003509011079415, 2447059985777227590, 83164038200838759780, 2826353783752411211145, 96054447135432681999180
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170754, 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(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-34*x+594*x^4-x^561*x^5) )); // G. C. Greubel, Apr 30 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(561*t^4-33*t^3-33*t^2 - 33*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{33, 33, 33, -561}, {1, 35, 1190, 40460}, 20] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 561, -33}] (* 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)/(561*t^4-33*t^3 - 33*t^2-33*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-34*x+594*x^4-561*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(561*t^4 - 33*t^3 - 33*t^2 - 33*t + 1).
a(n) = -561*a(n-4) + 33*Sum_{k=1..3} a(n-k). - Wesley Ivan Hurt, May 05 2021

A163219 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 36, 1260, 44100, 1542870, 53978400, 1888472880, 66069561600, 2311490430270, 80869130653500, 2829263840578980, 98983800307381500, 3463018394666864670, 121156152466965222600, 4238733846520797445080, 148295107229819712107400
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170755, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A154638, A170755.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-35*x+629*x^4-595*x^5) )); // G. C. Greubel, Apr 30 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(595*t^4-34*t^3-34*t^2 - 34*t+1), {t, 0, 20}], t] (* or *) Join[{1}, LinearRecurrence[{34, 34, 34, -595}, {36, 1260, 44100, 1542870}, 20]] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 595, -34}] (* 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)/(595*t^4-34*t^3 - 34*t^2-34*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-35*x+629*x^4-595*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(595*t^4 - 34*t^3 - 34*t^2 - 34*t + 1).
a(n) = -595*a(n-4) + 34*Sum_{k=1..3} a(n-k). - Wesley Ivan Hurt, May 05 2021

A163220 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 37, 1332, 47952, 1725606, 62097840, 2234659770, 80416702800, 2893883982570, 104139615440700, 3747579228757350, 134860782963557700, 4853114416362432150, 174644689291688511000, 6284782282271390399250
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-36*x+665*x^4-630*x^5) )); // G. C. Greubel, Apr 30 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(630*t^4-35*t^3-35*t^2 - 35*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{35, 35, 35, -630}, {1, 37, 1332, 47952}, 20] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 630, -35}] (* 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)/(630*t^4-35*t^3 - 35*t^2-35*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-36*x+665*x^4-630*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).
a(n) = -630*a(n-4) + 35*Sum_{k=1..3} a(n-k). - Wesley Ivan Hurt, May 05 2021

A163221 Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 38, 1406, 52022, 1924111, 71166096, 2632183848, 97355219328, 3600827035866, 133181923185576, 4925930761424952, 182192847843197736, 6738672428195210748, 249239784283952410080, 9218502714272560450272
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • GAP
    a:=[38,1406,52022,1924111];; for n in [5..20] do a[n]:=36*(a[n-1]+ a[n-2]+a[n-3]) -666*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)/(1-37*x+702*x^4-666*x^5) )); // G. C. Greubel, May 01 2019
  • Mathematica
    coxG[{4,666,-36}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 09 2015 *)
CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(666*t^4-36*t^3-36*t^2 - 36*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{36, 36, 36, -666}, {1, 38, 1406, 52022, 1924111}, 20] (* G. C. Greubel, Dec 11 2016; modified by Georg Fischer, Apr 08 2019 *)
  • PARI
    my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(666*t^4-36*t^3 - 36*t^2-36*t+1)) \\ G. C. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-37*x+702*x^4-666*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)/(666*t^4 - 36*t^3 - 36*t^2 - 36*t + 1).
a(n) = 36*a(n-1)+36*a(n-2)+36*a(n-3)-666*a(n-4). - Wesley Ivan Hurt, May 06 2021

A163222 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 39, 1482, 56316, 2139267, 81263988, 3086962281, 117263934684, 4454486050560, 169211838474861, 6427822638540342, 244172655087350379, 9275347010187982854, 352341101130365494992, 13384324210123816783899
Offset: 0

Views

Author

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

Keywords

Comments

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.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5) )); // G. C. Greubel, Apr 30 2019
  • Mathematica
    CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(703*t^4-37*t^3-37*t^2 - 37*t+1), {t,0,20}], t] (* or *) Join[{1}, LinearRecurrence[{37, 37, 37, -703}, {39, 1482, 56316, 2139267}, 20]] (* G. C. Greubel, Dec 11 2016 *)
coxG[{4, 703, -37}] (* 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)/(703*t^4-37*t^3 - 37*t^2-37*t+1)) \\ G. c. Greubel, Dec 11 2016
  • Sage
    ((1+x)*(1-x^4)/(1-38*x+740*x^4-703*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 30 2019

Formula

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(703*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).
a(n) = 37*a(n-1)+37*a(n-2)+37*a(n-3)-703*a(n-4). - Wesley Ivan Hurt, May 06 2021
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