A254600 -id:A254600 - cp's OEIS frontend

A255115 Number of n-length words on {0,1,2} in which 0 appears only in runs of length 2.

1, 2, 5, 12, 28, 66, 156, 368, 868, 2048, 4832, 11400, 26896, 63456, 149712, 353216, 833344, 1966112, 4638656, 10944000, 25820224, 60917760, 143723520, 339087488, 800010496, 1887468032, 4453111040, 10506243072, 24787422208, 58481066496, 137974619136

Offset: 0

Milan Janjic, Feb 14 2015

Apparently a(n) = A239333(n).

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (2,0,2).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 2,  a[2]== 5, a[n] == 2 a[n - 1] + 2 a[n - 3]}, a[n], {n, 0, 29}]

Vec(-(x^2+1)/(2*x^3+2*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 2*a(n+2) + 2*a(n) with n>1, a(0) = 1, a(1) = 2, a(2)=5.
G.f.: -(x^2+1) / (2*x^3+2*x-1). - Colin Barker, Feb 15 2015
a(n) = A052912(n)+A052912(n-2). - R. J. Mathar, Jun 18 2015

A255116 Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 3, 10, 33, 108, 354, 1161, 3807, 12483, 40932, 134217, 440100, 1443096, 4731939, 15516117, 50877639, 166828734, 547034553, 1793736576, 5881695930, 19286191449, 63239784075, 207364440015, 679951894392, 2229575035401, 7310818426248, 23972310961920

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (3,0,3).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 3,  a[2]== 10, a[n] == 3 a[n - 1] + 3 a[n - 3]}, a[n], {n, 0, 25}]
LinearRecurrence[{3,0,3},{1,3,10},30] (* Harvey P. Dale, Feb 20 2023 *)

Vec(-(x^2+1)/(3*x^3+3*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 3*a(n+2) + 3*a(n) with n>1, a(0) = 1, a(1) = 3, a(2) = 10.
G.f.: -(x^2+1) / (3*x^3+3*x-1). - Colin Barker, Feb 15 2015
a(n) = A089978(n) + A089978(n-2). - R. J. Mathar, Aug 04 2019

A255117 Number of n-length words on {0,1,2,3,4} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 4, 17, 72, 304, 1284, 5424, 22912, 96784, 408832, 1726976, 7295040, 30815488, 130169856, 549859584, 2322700288, 9811480576, 41445360640, 175072243712, 739534897152, 3123921031168, 13195973099520, 55742031986688, 235463812071424, 994639140683776

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
Index entries for linear recurrences with constant coefficients, signature (4,0,4).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 4,  a[2]== 17, a[n] == 4 a[n - 1] + 4 a[n - 3]}, a[n], {n, 0, 25}]

Vec(-(x^2+1)/(4*x^3+4*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 4*a(n+2) + 4*a(n) with n>1, a(0) = 1, a(1) = 4, a(2) = 17.
G.f.: -(x^2+1) / (4*x^3+4*x-1). - Colin Barker, Feb 15 2015
a(n) = A089979(n) + A089979(n-2). - R. J. Mathar, Aug 04 2019

A255118 Number of n-length words on {0,1,2,3,4,5} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 5, 26, 135, 700, 3630, 18825, 97625, 506275, 2625500, 13615625, 70609500, 366175000, 1898953125, 9847813125, 51069940625, 264844468750, 1373461409375, 7122656750000, 36937506093750, 191554837515625, 993387471328125, 5151624887109375, 26715898623125000

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10
Index entries for linear recurrences with constant coefficients, signature (5,0,5).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 5,  a[2]== 26, a[n] == 5 a[n - 1] + 5 a[n - 3]}, a[n], {n, 0, 20}]

Vec(-(x^2+1)/(5*x^3+5*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 5*a(n+2) + 5*a(n) with n>1, a(0) = 1, a(1) = 5, a(2) = 26.

G.f.: -(x^2+1) / (5*x^3+5*x-1). - Colin Barker, Feb 15 2015

A015591 Expansion of x/(1 - 10x - 9x^2).

Original entry on oeis.org

0, 1, 10, 109, 1180, 12781, 138430, 1499329, 16239160, 175885561, 1905008050, 20633050549, 223475577940, 2420453234341, 26215812544870, 283942204557769, 3075364358481520, 33309123425835121, 360769513484684890, 3907477245679364989, 42321698078155813900

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 2, 1, 4, 4, 2, 48, 8, 1, 4, 10, 4, 84, 48, 4, 16, 272, 2, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,9).

Cf. A254600.

[n le 2 select n-1 else 10*Self(n-1) + 9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

Join[{a=0,b=1},Table[c=10*b+9*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011 *)
LinearRecurrence[{10, 9}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)
Table[(-3 I)^(n - 1)*ChebyshevU[n - 1, 5 I/3], {n, 0, 30}] (* G. C. Greubel, Feb 13 2021 *)

[lucas_number1(n,10,-9) for n in range(0, 18)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*a(n-1) + 9*a(n-2).

a(n) = (-3*i)^(n-1) * ChebyshevU(n-1, -5*i/3). - G. C. Greubel, Feb 13 2021

Extended by T. D. Noe, May 23 2011

A255119 Number of n-length words on {0,1,2,3,4,5,6} in which 0 appears only in runs of length 2.

Original entry on oeis.org

1, 6, 37, 228, 1404, 8646, 53244, 327888, 2019204, 12434688, 76575456, 471567960, 2904015888, 17883548064, 110130696144, 678208272192, 4176550921536, 25720089706080, 158389787869632, 975398032747008, 6006708734718528, 36990591135528960

Offset: 0

Milan Janjic, Feb 14 2015

Colin Barker, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3. example 10.
Index entries for linear recurrences with constant coefficients, signature (6,0,6).

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

RecurrenceTable[{a[0] == 1, a[1] == 6,  a[2]== 37, a[n] == 6 a[n - 1] + 6 a[n - 3]}, a[n], {n, 0, 20}]
LinearRecurrence[{6,0,6},{1,6,37},30] (* Harvey P. Dale, Nov 06 2017 *)

Vec(-(x^2+1)/(6*x^3+6*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015

a(n+3) = 6*a(n+2) + 6*a(n) with n>1, a(0) = 1, a(1) = 6, a(2) = 37.

G.f.: -(x^2+1) / (6*x^3+6*x-1). - Colin Barker, Feb 15 2015

cp's OEIS Frontend

A255115 Number of n-length words on {0,1,2} in which 0 appears only in runs of length 2.

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A255116 Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.

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A255117 Number of n-length words on {0,1,2,3,4} in which 0 appears only in runs of length 2.

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A255118 Number of n-length words on {0,1,2,3,4,5} in which 0 appears only in runs of length 2.

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A015591 Expansion of x/(1 - 10*x - 9*x^2).

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A255119 Number of n-length words on {0,1,2,3,4,5,6} in which 0 appears only in runs of length 2.

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A015591 Expansion of x/(1 - 10x - 9x^2).