A287831 -id:A287831 - cp's OEIS frontend

A287825 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 1.

1, 10, 82, 674, 5540, 45538, 374316, 3076828, 25291120, 207889674, 1708825732, 14046322404, 115458919774, 949057110644, 7801124426174, 64124215108032, 527092600834054, 4332631742719370, 35613662169258228, 292739611493034596, 2406281042646218328

Offset: 0

David Nacin, Jun 02 2017

Index entries for linear recurrences with constant coefficients, signature (9, -4, -21, 9, 5).

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

LinearRecurrence[{9, -4, -21, 9, 5}, {1, 10, 82, 674, 5540, 45538}, 40]

def a(n):
    if n in [0, 1, 2, 3, 4, 5]:
        return [1, 10, 82, 674, 5540, 45538][n]
    return 9*a(n-1) - 4*a(n-2) - 21*a(n-3) + 9*a(n-4) + 5*a(n-5)

For n>5, a(n) = 9*a(n-1) - 4*a(n-2) - 21*a(n-3) + 9*a(n-4) + 5*a(n-5), a(0)=1, a(1)=10, a(2)=82, a(3)=674, a(4)=5540, a(5)=45538.

G.f.: (-1 - x + 4*x^2 + 3*x^3 - 3*x^4 - x^5)/(-1 + 9*x - 4*x^2 - 21*x^3 + 9*x^4 + 5*x^5).

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

cp's OEIS Frontend

A287825 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 1.

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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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Python

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