A287804 -id:A287804 - cp's OEIS frontend

A287827 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 3.

1, 10, 86, 742, 6404, 55274, 477082, 4117804, 35541714, 306768722, 2647791524, 22853698754, 197255539962, 1702558017644, 14695170558994, 126837403201602, 1094762853302164, 9449150445514434, 81557794797885642, 703944119701429084, 6075903902137709074

Offset: 0

David Nacin, Jun 02 2017

Index entries for linear recurrences with constant coefficients, signature (9, -1, -20, 10).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

LinearRecurrence[{9, -1, -20, 10}, {1, 10, 86, 742, 6404}, 30]

def a(n):
 if n in [0, 1, 2, 3, 4]:
  return [1, 10, 86, 742, 6404][n]
 return 9*a(n-1)-a(n-2)-20*a(n-3)+10*a(n-4)

For n>4, a(n) = 9*a(n-1) - a(n-2) - 20*a(n-3) + 10*a(n-4), a(0)=1, a(1)=10, a(2)=86, a(3)=742, a(4)=6404.

G.f.: (-1 - x + 3*x^2 + 2*x^3 - 2*x^4)/(-1 + 9*x - x^2 - 20*x^3 + 10*x^4).

A287828 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 4.

Original entry on oeis.org

1, 10, 88, 776, 6844, 60364, 532412, 4695892, 41417932, 365307620, 3222026092, 28418383780, 250651147340, 2210751960772, 19498910274028, 171981076403492, 1516879160180620, 13378927697789188, 118002614210453804, 1040787219651555556, 9179779989094951372

Offset: 0

David Nacin, Jun 02 2017

Index entries for linear recurrences with constant coefficients, signature (9, 0, -14).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

LinearRecurrence[{9, 0, -14}, {1, 10, 88, 776}, 30]

def a(n):
 if n in [0, 1, 2, 3]:
  return [1, 10, 88, 776][n]
 return 9*a(n-1)-14*a(n-3)

For n>3, a(n) = 9*a(n-1) - 14*a(n-3), a(0)=1, a(1)=10, a(2)=88, a(3)=776.

G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 9*x + 14*x^3).

A287829 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 6.

Original entry on oeis.org

1, 10, 92, 848, 7816, 72040, 663992, 6120008, 56408056, 519912520, 4792028792, 44168084168, 407096815096, 3752207504200, 34584061167992, 318760965520328, 2938016812018936, 27079673239211080, 249593092776937592, 2300497181470860488, 21203660818791619576

Offset: 0

David Nacin, Jun 02 2017

In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

Index entries for linear recurrences with constant coefficients, signature (9,2).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

Mathematica
```
LinearRecurrence[{9, 2}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+2*a(n-2)

a(n) = 9*a(n-1) + 2*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 2*x^2).
a(n) = ((1 - 11/sqrt(89))/2)*((9 - sqrt(89))/2)^n + ((1 + 11/sqrt(89))/2)*((9 + sqrt(89))/2)^n.
a(n) = A015579(n)+A015579(n+1). - R. J. Mathar, Oct 20 2019

A287830 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 7.

Original entry on oeis.org

1, 10, 94, 886, 8350, 78694, 741646, 6989590, 65872894, 620814406, 5850821230, 55140648694, 519669123166, 4897584703270, 46156938822094, 435002788211926, 4099652849195710, 38636886795609094, 364130592557264686, 3431722880197818550, 32342028292009425694

Offset: 0

David Nacin, Jun 02 2017

In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

Index entries for linear recurrences with constant coefficients, signature (9,4).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

Mathematica
```
LinearRecurrence[{9, 4}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+4*a(n-2)

a(n) = 9*a(n-1) + 4*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 4*x^2).
a(n) = ((1 - 11/sqrt(97))/2)*((9 - sqrt(97))/2)^n + ((1 + 11/sqrt(97))/2)*((9 + sqrt(97))/2)^n.
a(n) = A015580(n)+A015580(n+1). - R. J. Mathar, Oct 20 2019

A287832 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 11, 101, 929, 8545, 78599, 722973, 6650087, 61169169, 562649373, 5175390189, 47604538285, 437878494689, 4027716327495, 37047945974857, 340776308298291, 3134546038698889, 28832341420057365, 265207115001514409, 2439441626426418609, 22438596523731989473

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (11,-14,-28,39,9,-10).

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

LinearRecurrence[{11, -14, -28, 39, 9, -10}, {1, 11, 101, 929, 8545, 78599, 722973}, 20]

def a(n):
 if n in [0,1,2,3,4,5,6]:
  return [1, 11, 101, 929, 8545, 78599, 722973][n]
 return 11*a(n-1) - 14*a(n-2) - 28*a(n-3) + 39*a(n-4) + 9*a(n-5) - 10*a(n-6)

For n>6, a(n) = 11*a(n-1) - 14*a(n-2) - 28*a(n-3) + 39*a(n-4) + 9*a(n-5) - 10*a(n-6), a(0)=1, a(1)=11, a(2)=101, a(3)=929, a(4)=8545, a(5)=78599, a(6)=722973.

G.f.: (1 - 6*x^2 + 9*x^4 - 2*x^6)/(1 - 11*x + 14*x^2 + 28*x^3 - 39*x^4 - 9*x^5 + 10*x^6).

A287833 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 11, 103, 967, 9079, 85243, 800351, 7514541, 70554457, 662439857, 6219685951, 58396989455, 548292695881, 5147951686649, 48334414751849, 453814602701801, 4260891430727991, 40005754941255473, 375616336261903907, 3526683405274793053, 33112233522155404139

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,-2,-37,16,19,1).

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

LinearRecurrence[{10, -2, -37, 16, 19, 1}, {1, 11, 103, 967, 9079, 85243}, 20]

def a(n):
 if n in [0,1,2,3,4,5]:
  return [1, 11, 103, 967, 9079, 85243][n]
 return 10*a(n-1) - 2*a(n-2) - 37*a(n-3) + 16*a(n-4) + 19*a(n-5) + a(n-6)

a(n) = 10*a(n-1) - 2*a(n-2) - 37*a(n-3) + 16*a(n-4) + 19*a(n-5) + a(n-6), a(0)=1, a(1)=11, a(2)=103, a(3)=967, a(4)=9079, a(5)=85243.

G.f.: (-1 - x + 5*x^2 + 4*x^3 - 6*x^4 - 3*x^5)/(-1 + 10*x - 2*x^2 - 37*x^3 + 16*x^4 + 19*x^5 + x^6).

A287834 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 3.

Original entry on oeis.org

1, 11, 105, 1005, 9621, 92105, 881753, 8441329, 80811789, 773639469, 7406320733, 70903294113, 678781988705, 6498216958121, 62209699634757, 595555173609653, 5701457600593525, 54582044135967257, 522532964509030377, 5002390498942001761, 47889630709552579709

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,-2,-21,10).

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

LinearRecurrence[{10, -2, -21, 10}, {1, 11, 105, 1005, 9621}, 20]

def a(n):
 if n in [0,1,2,3,4]:
  return [1, 11, 105, 1005, 9621][n]
 return 10*a(n-1) - 2*a(n-2) - 21*a(n-3) + 10*a(n-4)

a(n) = 10*a(n-1) - 2*a(n-2) - 21*a(n-3) + 10*a(n-4), a(0)=1, a(1)=11, a(2)=105, a(3)=1005, a(4)=9621.

G.f.: (-1 - x + 3*x^2 + 2*x^3 - 2*x^4)/(-1 + 10*x - 2*x^2 - 21*x^3 + 10*x^4).

A287835 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 4.

Original entry on oeis.org

1, 11, 107, 1043, 10169, 99149, 966719, 9425675, 91901945, 896059709, 8736735695, 85184670011, 830565128489, 8098152315149, 78958372642847, 769857662314475, 7506244118089817, 73187166301583837, 713587411625345903, 6957599532298617755, 67837787583138657929

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,-1,-14)

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

LinearRecurrence[{10, -1, -14}, {1, 11, 107, 1043}, 20]

def a(n):
 if n in [0,1,2,3]:
  return [1, 11, 107, 1043][n]
 return 10*a(n-1) - a(n-2) - 14*a(n-3)

For n>3, a(n) = 10*a(n-1) - a(n-2) - 14*a(n-3), a(0)=1, a(1)=11, a(2)=107, a(3)=1043.

G.f.: (1 + x - 2 x^2 - 2 x^3)/(1 - 10 x + x^2 + 14 x^3).

A287836 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 11, 109, 1081, 10721, 106329, 1054553, 10458881, 103729441, 1028771337, 10203182953, 101193470929, 1003620008177, 9953736259545, 98719500126905, 979083577381409, 9710388021269185, 96306012787788969, 955147011506293513, 9472989143467878769, 93951530216004879761

Offset: 0

David Nacin, Jun 07 2017

Index entries for linear recurrences with constant coefficients, signature (10,1,-18).

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

LinearRecurrence[{10, 1, -18}, {1, 11, 109, 1081}, 20]

def a(n):
 if n in [0,1,2,3]:
  return [1, 11, 109, 1081][n]
 return 10*a(n-1) + a(n-2) - 18*a(n-3)

For n>3, a(n) = 10*a(n-1) + a(n-2) - 18*a(n-3), a(0)=1, a(1)=11, a(2)=109, a(3)=1081.

G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 10*x - x^2 + 18*x^3).

A287837 Number of words over the alphabet {0,1,...,10} such that no two consecutive terms have distance 7.

Original entry on oeis.org

1, 11, 113, 1163, 11969, 123179, 1267697, 13046507, 134268161, 1381821131, 14221015793, 146355621323, 1506219260609, 15501259470059, 159531252482417, 1641816303234347, 16896756789790721, 173893016807610251, 1789620438445474673, 18417883434877577483

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).

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

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

LinearRecurrence[{10, 3}, {1, 11, 113}, 20]

def a(n):
 if n in [0,1,2]:
  return [1, 11, 113][n]
 return 10*a(n-1) + 3*a(n-2)

For n>2, a(n) = 10*a(n-1) + 3*a(n-2), a(0)=1, a(1)=11, a(2)=113.
G.f.: (-1 - x)/(-1 + 10*x + 3*x^2).
a(n) = A015588(n)+A015588(n+1). - R. J. Mathar, Oct 20 2019

cp's OEIS Frontend

A287827 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 3.

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A287828 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 4.

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A287829 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 6.

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A287830 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 7.

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A287832 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 1.

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A287833 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 2.

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A287834 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 3.

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A287835 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 4.

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