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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A170494 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^46 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080
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.
First disagreement at index 46: a(46) = 6189700196426901374495621110, A003947(46) = 6189700196426901374495621120.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

A003947 (g.f.: (1+x)/(1-4*x)).

Programs

Formula

G.f. (t^46 + 2*t^45 + 2*t^44 + 2*t^43 + 2*t^42 + 2*t^41 + 2*t^40 + 2*t^39 +
2*t^38 + 2*t^37 + 2*t^36 + 2*t^35 + 2*t^34 + 2*t^33 + 2*t^32 + 2*t^31 +
2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 +
2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*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)/(6*t^46 - 3*t^45 -
3*t^44 - 3*t^43 - 3*t^42 - 3*t^41 - 3*t^40 - 3*t^39 - 3*t^38 - 3*t^37 -
3*t^36 - 3*t^35 - 3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 - 3*t^30 - 3*t^29 -
3*t^28 - 3*t^27 - 3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 - 3*t^22 - 3*t^21 -
3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 -
3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4
- 3*t^3 - 3*t^2 - 3*t + 1)

A170542 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^47 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080
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

  • Mathematica
    With[{num=Total[2t^Range[46]]+t^47+1,den=Total[-3 t^Range[46]]+6t^47+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, May 26 2012 *)

Formula

G.f. (t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 + 2*t^42 + 2*t^41 + 2*t^40 +
2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*t^35 + 2*t^34 + 2*t^33 + 2*t^32 +
2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 +
2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*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)/(6*t^47 -
3*t^46 - 3*t^45 - 3*t^44 - 3*t^43 - 3*t^42 - 3*t^41 - 3*t^40 - 3*t^39 -
3*t^38 - 3*t^37 - 3*t^36 - 3*t^35 - 3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 -
3*t^30 - 3*t^29 - 3*t^28 - 3*t^27 - 3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 -
3*t^22 - 3*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 -
3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 -
3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1)

A170590 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^48 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080
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

Formula

G.f. (t^48 + 2*t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 + 2*t^42 + 2*t^41 +
2*t^40 + 2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*t^35 + 2*t^34 + 2*t^33 +
2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 +
2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*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)/(6*t^48
- 3*t^47 - 3*t^46 - 3*t^45 - 3*t^44 - 3*t^43 - 3*t^42 - 3*t^41 - 3*t^40
- 3*t^39 - 3*t^38 - 3*t^37 - 3*t^36 - 3*t^35 - 3*t^34 - 3*t^33 - 3*t^32
- 3*t^31 - 3*t^30 - 3*t^29 - 3*t^28 - 3*t^27 - 3*t^26 - 3*t^25 - 3*t^24
- 3*t^23 - 3*t^22 - 3*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16
- 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 -
3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1)

A170638 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^49 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080
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

Formula

G.f. (t^49 + 2*t^48 + 2*t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 + 2*t^42 +
2*t^41 + 2*t^40 + 2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*t^35 + 2*t^34 +
2*t^33 + 2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 +
2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 +
2*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)/(6*t^49 - 3*t^48 - 3*t^47 - 3*t^46 - 3*t^45 - 3*t^44 - 3*t^43 -
3*t^42 - 3*t^41 - 3*t^40 - 3*t^39 - 3*t^38 - 3*t^37 - 3*t^36 - 3*t^35 -
3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 - 3*t^30 - 3*t^29 - 3*t^28 - 3*t^27 -
3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 - 3*t^22 - 3*t^21 - 3*t^20 - 3*t^19 -
3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 -
3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 -
3*t + 1)

A287805 Number of quinary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 5, 19, 73, 281, 1083, 4175, 16097, 62065, 239307, 922711, 3557761, 13717913, 52893147, 203943935, 786361409, 3032030689, 11690820555, 45077144455, 173807214241, 670161078089, 2583988659867, 9963272432111, 38416111919777, 148123788152017, 571131629935179
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2)=19=25-6 sequences contain every combination except these six: 02,20,13,31,24,42.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{4, 1, -6}, {1, 5, 19, 73}, 40]
  • Python
    def a(n):
 if n in [0,1,2,3]:
  return [1,5,19,73][n]
 return 4*a(n-1)+a(n-2)-6*a(n-3)

Formula

For n>0, a(n) = 4*a(n-1) + a(n-2) - 6*a(n-3), a(1)=5, a(2)=19, a(3)=73.
G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 4*x - x^2 + 6*x^3).

A287806 Number of senary sequences of length n such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 6, 26, 114, 500, 2194, 9628, 42252, 185422, 813722, 3571010, 15671340, 68773514, 301811860, 1324498252, 5812546998, 25508302906, 111942925778, 491260382084, 2155891150146, 9461106209228, 41519967599596, 182209952129086, 799626506818554, 3509152727035810
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2)=26=36-10 sequences contain every combination except these ten: 01,10,12,21,23,32,34,43,45,54.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{5, -2, -3}, {1, 6, 26, 114}, 40]
  • Python
    def a(n):
 if n in [0, 1, 2, 3]:
  return [1, 6, 26, 114][n]
 return 5*a(n-1)-2*a(n-2)-3*a(n-3)

Formula

For n>3, a(n) = 5*a(n-1) - 2*a(n-2) - 3*a(n-3), a(1)=6, a(2)=26, a(3)=114.
G.f.: (1 + x - 2*x^2 - x^3)/(1 - 5*x + 2*x^2 + 3*x^3).

A287807 Number of senary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 6, 28, 132, 624, 2952, 13968, 66096, 312768, 1480032, 7003584, 33141312, 156826368, 742110336, 3511703808, 16617560832, 78635142144, 372105487872, 1760822074368, 8332299518976, 39428864667648, 186579390892032, 882903157346304, 4177942598725632
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2)=28=36-8 sequences contain every combination except these eight: 02,20,13,31,24,42,35,53.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{6, -6}, {1, 6, 28}, 40]
  • Python
    def a(n):
 if n in [0, 1, 2]:
  return [1, 6, 28][n]
 return 6*a(n-1)-6*a(n-2)

Formula

For n>2, a(n) = 6*a(n-1) - 6*a(n-2), a(1)=6, a(2)=28.
G.f.: (1 - 2*x^2)/(1 - 6*x + 6*x^2).

A287808 Number of septenary sequences of length n such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 7, 37, 197, 1049, 5587, 29757, 158491, 844153, 4496123, 23947233, 127547675, 679344041, 3618320227, 19271886609, 102645866251, 546712113769, 2911896468083, 15509334488577, 82605772190267, 439974623297369, 2343391557436483, 12481365289466289
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2)=37=49-12 sequences contain every combination except these twelve: 01,10,12,21,23,32,34,43,45,54,56,65.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{7, -8, -6, 6}, {1, 7, 37, 197, 1049}, 40]
  • Python
    def a(n):
 if n in [0,1,2,3,4]:
  return [1, 7, 37, 197, 1049][n]
 return 7*a(n-1)-8*a(n-2)-6*a(n-3)+6*a(n-4)

Formula

For n>4, a(n) = 7*a(n-1) - 8*a(n-2) - 6*a(n-3) + 6*a(n-4), a(1)=7, a(2)=37, a(3)=197, a(4)=1049.
G.f.: (1-4*x^2+2*x^4)/(1-7*x+8*x^2+6*x^3-6*x^4).

A287809 Number of septenary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 7, 39, 219, 1231, 6921, 38913, 218789, 1230147, 6916539, 38888455, 218651553, 1229375193, 6912200477, 38864063403, 218514412227, 1228604118319, 6907865088537, 38839687552689, 218377358251349, 1227833528067027, 6903532420748427, 38815326992539159
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2)=49-10=39 sequences contain every combination except these ten: 02,20,13,31,24,42,35,53,46,64.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{6, 0, -13, 6}, {1, 7, 39, 219, 1231}, 40]
  • Python
    def a(n):
 if n in [0, 1, 2, 3, 4]:
  return [1, 7, 39, 219, 1231][n]
 return 6*a(n-1)-13*a(n-3)+6*a(n-4)

Formula

For n>4, a(n) = 6*a(n-1) - 13*a(n-3) + 6*a(n-4), a(1)=7, a(2)=39, a(3)=219, a(4)=1231.
G.f.: (1 + x - 3*x^2 - 2*x^3 + 2*x^4)/(1 - 6*x + 13*x^3 - 6*x^4).

A287810 Number of septenary sequences of length n such that no two consecutive terms have distance 3.

Original entry on oeis.org

1, 7, 41, 241, 1417, 8333, 49005, 288193, 1694833, 9967141, 58615749, 344713305, 2027224169, 11921900829, 70111496093, 412318635697, 2424804301985, 14260029486677, 83861794865077, 493182755657289, 2900358033942041, 17056713010658765, 100308808541321741
Offset: 0

Views

Author

David Nacin, Jun 01 2017

Keywords

Examples

			For n=2 the a(2) = 49-8 = 41 sequences contain every combination except these eight: 03, 30, 14, 41, 25, 52, 36, 63.

Links

Crossrefs

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473.
Cf. A287804-A287819.

Programs

  • Mathematica
    LinearRecurrence[{6, 1, -10}, {1, 7, 41, 241}, 40]
  • Python
    def a(n):
 if n in [0, 1, 2, 3]:
  return [1, 7, 41, 241][n]
 return 6*a(n-1)+a(n-2)-10*a(n-3)

Formula

For n>3, a(n) = 6*a(n-1) + a(n-2) - 10*a(n-3), a(0)=1, a(1)=7, a(2)=41, a(3)=241.
G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 6*x - x^2 + 10*x^3).
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