A287819 -id:A287819 - cp's OEIS frontend

A287804 Number of quinary sequences of length n such that no two consecutive terms have distance 1.

1, 5, 17, 59, 205, 713, 2481, 8635, 30057, 104629, 364225, 1267923, 4413861, 15365465, 53490097, 186209299, 648230545, 2256616133, 7855718641, 27347281995, 95201200637, 331413874569, 1153716087665, 4016309864843, 13981555011321, 48672509644725

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2)=17=25-8 sequences contain every combination except these eight: 01,10,12,21,23,32,34,43.

Index entries for linear recurrences with constant coefficients, signature (5,-5,-1).

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

LinearRecurrence[{5, -5, -1}, {1, 5, 17}, 50]

def a(n):
    if n in [0,1,2]:
        return [1,5,17][n]
    return 5*a(n-1)-5*a(n-2)-a(n-3)

a(n) = 5*a(n-1) - 5a(n-2) - a(n-3), a(0)=1, a(1)=5, a(2)=17.

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

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

Original entry on oeis.org

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).

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

Original entry on oeis.org

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.

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).

cp's OEIS Frontend

A287804 Number of quinary sequences of length n such that no two consecutive terms have distance 1.

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

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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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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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PARI

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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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