A287825 -id:A287825 - cp's OEIS frontend

A287839 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 9.

1, 11, 117, 1247, 13289, 141619, 1509213, 16083463, 171399121, 1826575451, 19465548357, 207441511727, 2210673955769, 23558830139779, 251063019088173, 2675542001860183, 28512861152219041, 303857405535211691, 3238164083417650197, 34508642672922983807

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 (10,7).

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

a:=proc(n) option remember; if n=0 then 1 elif n=1 then 11 elif n=2 then 117 else 10*a(n-1)+7*a(n-2); fi; end: seq(a(n), n=0..30); # Wesley Ivan Hurt, Nov 25 2017

LinearRecurrence[{10, 7}, {1, 11, 117}, 20]

Vec((1 + x) / (1 - 10*x - 7*x^2) + O(x^30)) \\ Colin Barker, Nov 25 2017

def a(n):
 if n in [0,1,2]:
  return [1, 11, 117][n]
 return 10*a(n-1) + 7*a(n-2)

For n>2, a(n) = 10*a(n-1) + 7*a(n-2), a(0)=1, a(1)=11, a(2)=117.
G.f.: (-1 - x)/(-1 + 10 x + 7 x^2).
a(n) = (((5-4*sqrt(2))^n*(-3+2*sqrt(2)) + (3+2*sqrt(2))*(5+4*sqrt(2))^n)) / (4*sqrt(2)). - Colin Barker, Nov 25 2017

A287831 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 10, 96, 924, 8892, 85572, 823500, 7924932, 76265388, 733938084, 7063035084, 67970944260, 654116708844, 6294876045156, 60578584659468, 582976518206148, 5610260171812140, 53990200655546148, 519573366930788172, 5000101506310370436, 48118353758378062956

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 (9,6).

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

Mathematica
```
LinearRecurrence[{9, 6}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+6*a(n-2)

a(n) = 9*a(n-1) + 6*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 6*x^2).
a(n) = ((1 - 11/sqrt(105))/2)*((9 - sqrt(105))/2)^n + ((1 + 11/sqrt(105))/2)*((9 + sqrt(105))/2)^n.

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

A287826 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 10, 84, 708, 5968, 50308, 424080, 3574860, 30134944, 254028100, 2141377008, 18051134892, 152165391616, 1282706408548, 10812811724688, 91148603152524, 768354066287200, 6476983198439812, 54598931916359472, 460251829451302764, 3879778213203474880

Offset: 0

David Nacin, Jun 02 2017

Index entries for linear recurrences with constant coefficients, signature (10, -13, -2).

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

LinearRecurrence[{10, -13, -2}, {1, 10, 84}, 40]

def a(n):
 if n in [0, 1, 2]:
  return [1, 10, 84][n]
 return 10*a(n-1)-13*a(n-2)-2*a(n-3)

a(n) = 10*a(n-1) - 13*a(n-2) - 2a(n-3), a(0)=1, a(1)=10, a(2)=84.

G.f.: (1 - 3 x^2)/(1 - 10 x + 13 x^2 + 2 x^3).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A287839 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 9.

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A287831 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.

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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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A287826 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 2.

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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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