A126473 -id:A126473 - cp's OEIS frontend

A287813 Number of octonary sequences of length n such that no two consecutive terms have distance 2.

1, 8, 52, 340, 2224, 14548, 95164, 622504, 4072036, 26636740, 174241072, 1139777284, 7455717772, 48770692552, 319027694548, 2086881784180, 13651089405616, 89296980486772, 584125595190556, 3820988224873576, 24994540788543364, 163498820845182820

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 64 - 12 = 52 sequences contain every combination except these twelve: 02,20,13,31,24,42,35,53,46,64,57,75.

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

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

LinearRecurrence[{7, -3}, {1, 8, 52}, 40]

def a(n):
 if n in [0, 1, 2]:
  return [1, 8, 52][n]
 return 7*a(n-1)-3*a(n-2)

For n>2, a(n) = 7*a(n-1) - 3*a(n-2), a(0)=1, a(1)=8, a(2)=52.
G.f.: (1 + x - x^2)/(1 - 7 x + 3 x^2).
a(n) = A190972(n) + A190972(n+1) - A190972(n-1). - R. J. Mathar, Oct 20 2019

A287814 Number of octonary sequences of length n such that no two consecutive terms have distance 3.

Original entry on oeis.org

1, 8, 54, 366, 2482, 16834, 114178, 774426, 5252642, 35626714, 241642738, 1638972746, 11116542082, 75399367194, 511405842898, 3468675479466, 23526734684322, 159573084361274, 1082324835734258, 7341006503296586, 49791314679463362, 337715954398900954

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 64 - 10 = 54 sequences contain every combination except these ten: 03,30,14,41,25,52,36,63,47,74.

Index entries for linear recurrences with constant coefficients, signature (7, 0, -10).

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

LinearRecurrence[{7, 0, -10}, {1, 8, 54, 366}, 40]

def a(n):
 if n in [0, 1, 2, 3]:
  return [1, 8, 54, 366][n]
 return 7*a(n-1)-10*a(n-3)

For n>3, a(n) = 7*a(n-1) - 10*a(n-3), a(0)=1, a(1)=8, a(2)=54, a(3)=366.

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

A287815 Number of octonary sequences of length n such that no two consecutive terms have distance 7.

Original entry on oeis.org

1, 8, 62, 482, 3746, 29114, 226274, 1758602, 13667858, 106226618, 825593474, 6416514026, 49869159026, 387583197338, 3012297335522, 23411580532682, 181954847741906, 1414153417389434, 10990803008177474, 85420541561578922, 663888608980117298, 5159743512230294618

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 64 - 2 = 62 sequences contain every combination except these two: 07,70.

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

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

Mathematica
```
LinearRecurrence[{7, 6}, {1, 8}, 40]
```

def a(n):
 if n in [0, 1]:
  return [1, 8][n]
 return 7*a(n-1)+6*a(n-2)

a(n) = 7*a(n-1) + 6*a(n-2), a(0)=1, a(1)=8.
G.f.: (-1 - x)/(-1 + 7 x + 6 x^2).
a(n) = A015564(n)+A015564(n+1). - R. J. Mathar, Oct 20 2019

A287816 Number of nonary sequences of length n such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 9, 65, 471, 3413, 24733, 179233, 1298853, 9412437, 68209395, 494295113, 3582023557, 25957960001, 188110345129, 1363185009337, 9878634630295, 71587804656589, 518777540353453, 3759441118026705, 27243657291488469, 197427447142906157, 1430703538380753875

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 81 - 16 = 65 sequences contain every combination except these sixteen: 01,10,12,21,23,32,34,43,45,54,56,65,67,76,78,87.

Index entries for linear recurrences with constant coefficients, signature (9, -11, -15, 19, 1).

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

LinearRecurrence[{9, -11, -15, 19, 1}, {1, 9, 65 , 471, 3413}, 40]

def a(n):
 if n in [0, 1, 2, 3, 4]:
  return [1, 9, 65 , 471, 3413][n]
 return 9*a(n-1)-11*a(n-2)-15*a(n-3)+19*a(n-4)+a(n-5)

a(n) = 9*a(n-1) - 11*a(n-2) - 15*a(n-3) + 19*a(n-4) + a(n-5), a(0)=1, a(1)=9, a(2)=65, a(3)=471, a(4)=3413.

G.f: (-1 + 5 x^2 - 5 x^4)/(-1 + 9 x - 11 x^2 - 15 x^3 + 19 x^4 + x^5).

A287817 Number of nonary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 9, 67, 501, 3747, 28025, 209609, 1567743, 11725731, 87701095, 655949055, 4906086571, 36694443381, 274451368893, 2052723708275, 15353082914309, 114831408642039, 858866749063989, 6423783365292409, 48045861327359751, 359352839194448551, 2687733333725785179

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 81 - 14 = 67 sequences contain every combination except these fourteen: 02,20,13,31,24,42,35,53,46,64,57,75,68,86.

Index entries for linear recurrences with constant coefficients, signature (8, -1, -23, 10, 1).

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

LinearRecurrence[{8, -1, -23, 10, 1}, {1, 9, 67 , 501, 3747}, 40]

def a(n):
 if n in [0, 1, 2, 3, 4]:
  return [1, 9, 67 , 501, 3747][n]
 return 8*a(n-1)-a(n-2)-23*a(n-3)+10*a(n-4)+a(n-5)

a(n) = 8*a(n-1) - 1*a(n-2) - 23*a(n-3) + 10*a(n-4) + a(n-5), a(0)=1, a(1)=9, a(2)=67, a(3)=501, a(4)=3747.

G.f: (-1 - x + 4 x^2 + 3 x^3 - 3 x^4)/(-1 + 8 x - x^2 - 23 x^3 + 10 x^4 + x^5).

A287818 Number of nonary sequences of length n such that no two consecutive terms have distance 3.

Original entry on oeis.org

1, 9, 69, 531, 4089, 31491, 242529, 1867851, 14385369, 110789811, 853254609, 6571393371, 50609994249, 389776014531, 3001884188289, 23119197549291, 178053936060729, 1371293449053651, 10561101680875569, 81336980637343611, 626421808927336809, 4824426473972595171

Offset: 0

David Nacin, Jun 02 2017

			For n=2 the a(2) = 81 - 12 = 69 sequences contain every combination except these twelve: 03,30,14,41,25,52,36,63,47,74,58,85.

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

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

LinearRecurrence[{9, -10}, {1, 9, 69}, 40]

def a(n):
 if n in [0, 1, 2]:
  return [1, 9, 69][n]
 return 9*a(n-1)-10*a(n-2)

For n>2, a(n) = 9*a(n-1) - 10*a(n-2), a(0)=1, a(1)=9, a(2)=69.
G.f.: (1 - 2 x^2)/(1 - 9 x + 10 x^2).
For n>0, a(n)=(1/5)(3 - 18/sqrt(41))*((9 - sqrt(41))/2)^n + (1/5)(3 + 18/sqrt(41))*((9 + sqrt(41))/2)^n.
a(n) = A178869(n+1)-2*A178869(n-1). - R. J. Mathar, Oct 20 2019

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

cp's OEIS Frontend

A287813 Number of octonary sequences of length n such that no two consecutive terms have distance 2.

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A287814 Number of octonary sequences of length n such that no two consecutive terms have distance 3.

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A287815 Number of octonary sequences of length n such that no two consecutive terms have distance 7.

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A287816 Number of nonary sequences of length n such that no two consecutive terms have distance 1.

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A287817 Number of nonary sequences of length n such that no two consecutive terms have distance 2.

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A287818 Number of nonary sequences of length n such that no two consecutive terms have distance 3.

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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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