A126473 -id:A126473 - cp's OEIS frontend

A254662 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,4}.

1, 8, 59, 437, 3236, 23963, 177449, 1314032, 9730571, 72056093, 533584364, 3951258827, 29259564881, 216670730648, 1604473809179, 11881328856197, 87982723420916, 651523050515003, 4824609523867769, 35726835818619392, 264561679301939051, 1959112262569431533

Offset: 0

Milan Janjic, Feb 04 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,3).

Cf. A015559, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 8^n else 7*Self(n)+3*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 3 a[n - 2]}, a[n], {n, 0, 20}]

Vec((1+x)/(1-7*x-3*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

G.f.: (1 + x)/(1 - 7*x -3*x^2).
a(n) = 7*a(n-1) + 3*a(n-2) with n>1, a(0) = 1, a(1) = 8.
a(n) = (2^(-1-n) * ((7-sqrt(61))^n * (-9+sqrt(61)) + (7+sqrt(61))^n * (9+sqrt(61)))) / sqrt(61). - Colin Barker, Sep 08 2016

A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 7, 45, 291, 1881, 12159, 78597, 508059, 3284145, 21229047, 137226717, 887047443, 5733964809, 37064931183, 239591481525, 1548743682699, 10011236540769, 64713650292711, 418315611378573, 2704034619149571, 17479154549033145, 112987031151647583

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2) = 49-4 = 45 sequences contain every combination except these four: 05, 50, 16, 61.

Index entries for linear recurrences with constant coefficients, signature (6, 3).

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

Cf. A287804-A287819.

Mathematica
```
LinearRecurrence[{6, 3}, {1,7}, 40]
```

def a(n):
 if n in [0, 1]:
  return [1, 7][n]
 return 6*a(n-1)-3*a(n-2)

a(n) = 6*a(n-1) + 3*a(n-2), a(0)=1, a(1)=7.
G.f.: (1 + x)/(1 - 6*x - 3*x^2).
a(n) = A090018(n-1)+A090018(n). - R. J. Mathar, Oct 20 2019

A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 11, 115, 1205, 12625, 132275, 1385875, 14520125, 152130625, 1593906875, 16699721875, 174966753125, 1833166140625, 19206495171875, 201230782421875, 2108340300078125, 22089556912890625, 231437270629296875, 2424820490857421875, 25405391261720703125

Offset: 0

David Nacin, Jun 07 2017

In general, the number of sequences on {0,1,...,10} such that no two consecutive terms have distance 6+k for k in {0,1,2,3,4} has generating function (-1 - x)/(-1 + 10*x + (2*k+1)*x^2).

Colin Barker, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,5).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 5}, {1, 11, 115}, 20]

Vec((1 + x) / (1 - 10*x - 5*x^2) + O(x^40)) \\ Colin Barker, Nov 25 2017

def a(n):
 if n in [0,1,2]:
  return [1, 11, 115][n]
 return 10*a(n-1) + 5*a(n-2)

For n > 2, a(n) = 10*a(n-1) + 5*a(n-2), a(0)=1, a(1)=11, a(2)=115.
G.f.: (-1 - x)/(-1 + 10*x + 5*x^2).
a(n) = (((5-sqrt(30))^n*(-6+sqrt(30)) + (5+sqrt(30))^n*(6+sqrt(30)))) / (2*sqrt(30)). - Colin Barker, Nov 25 2017

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

David Nacin, Jun 01 2017

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

Index entries for linear recurrences with constant coefficients, signature (4, 1, -6).

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

LinearRecurrence[{4, 1, -6}, {1, 5, 19, 73}, 40]

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)

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

David Nacin, Jun 01 2017

			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.

Index entries for linear recurrences with constant coefficients, signature (5, -2, -3).

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

LinearRecurrence[{5, -2, -3}, {1, 6, 26, 114}, 40]

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)

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

David Nacin, Jun 01 2017

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

Index entries for linear recurrences with constant coefficients, signature (6, -6).

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

LinearRecurrence[{6, -6}, {1, 6, 28}, 40]

def a(n):
 if n in [0, 1, 2]:
  return [1, 6, 28][n]
 return 6*a(n-1)-6*a(n-2)

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

David Nacin, Jun 01 2017

			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.

Index entries for linear recurrences with constant coefficients, signature (7, -8, -6, 6).

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

LinearRecurrence[{7, -8, -6, 6}, {1, 7, 37, 197, 1049}, 40]

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)

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

David Nacin, Jun 01 2017

			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.

Index entries for linear recurrences with constant coefficients, signature (6, 0, -13, 6).

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

LinearRecurrence[{6, 0, -13, 6}, {1, 7, 39, 219, 1231}, 40]

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)

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

David Nacin, Jun 01 2017

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

Index entries for linear recurrences with constant coefficients, signature (6, 1, -10).

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

Cf. A287804-A287819.

LinearRecurrence[{6, 1, -10}, {1, 7, 41, 241}, 40]

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)

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).

A287812 Number of octonary sequences of length n such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 8, 50, 314, 1972, 12386, 77796, 488636, 3069120, 19277130, 121079578, 760500364, 4776699874, 30002433636, 188445170924, 1183623397912, 7434334035874, 46695023649050, 293291264969380, 1842161313673506, 11570608166423524, 72674945645197500

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 64 - 14 = 50 sequences contain every combination except these fourteen: 01,10,12,21,23,32,34,43,45,54,56,65,67,76.

Index entries for linear recurrences with constant coefficients, signature (7, -3, -10, 3).

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

LinearRecurrence[{7, -3, -10, 3}, {1, 8, 50, 314, 1972}, 40]

def a(n):
 if n in [0, 1, 2, 3, 4]:
  return [1, 8, 50, 314, 1972][n]
 return 7*a(n-1)-3*a(n-2)-10*a(n-3)+3*a(n-4)

For n>4, a(n) = 7*a(n-1) - 3*a(n-2) - 10*a(n-3) + 3*a(n-2), a(0)=1, a(1)=8, a(2)=50, a(3)=314, a(4)=1972.

G.f.: (-1 - x + 3 x^2 + 2 x^3 - x^4)/(-1 + 7 x - 3 x^2 - 10 x^3 + 3 x^4).

cp's OEIS Frontend

A254662 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,4}.

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A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

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A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

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

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

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

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Mathematica

Python

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

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Author

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Mathematica

Python

Formula

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