A190560 -id:A190560 - cp's OEIS frontend

A254657 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2}.

1, 9, 78, 678, 5892, 51204, 444984, 3867096, 33606672, 292055952, 2538087648, 22057036896, 191684821056, 1665820789824, 14476675244928, 125808326698368, 1093326665056512, 9501463280642304, 82571666235477504, 717582109567673856, 6236086873954255872

Offset: 0

Milan Janjic, Feb 04 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,6).

Cf. A055099, A126473, A126501, A126528, A190560, A254598, A254602.

[n le 1 select 9^n else 8*Self(n)+6*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 6 a[n - 2]}, a[n], {n, 0, 20}]

Vec((1+x)/(1-8*x-6*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

G.f.: (1 + x)/(1 - 8*x - 6*x^2).
a(n) = 8*a(n-1) + 6*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = (((4-sqrt(22))^n*(-5+sqrt(22)) + (4+sqrt(22))^n*(5+sqrt(22))))/(2*sqrt(22)). - Colin Barker, Nov 16 2016

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, -1, 3, 5, 1, -1, -1, 10, 7, 1, 0, -6, 7, 21, 9, 1, 1, -6, -10, 31, 36, 11, 1, 1, 1, -29, 7, 79, 55, 13, 1, 0, 9, -24, -63, 81, 159, 78, 15, 1, -1, 9, 15, -123, -54, 264, 279, 105, 17, 1, -1, -1, 57, -69, -321, 132, 624, 447, 136, 19, 1, 0, -12, 50, 126, -459, -507, 741, 1245, 671, 171, 21, 1, 1, -12, -20, 302, -81, -1419, -258, 2163, 2227, 959, 210, 23, 1

Offset: 1

Ilya Gutkovskiy, Apr 08 2016

The polynomials Q_n(x) have generating function G(x,t) = t/(1 - (x + 1)*t - (x - 1)*t^2) = t + (x + 1)*t^2 + x*(x + 3)*t^3 + (x^3 + 5*x^2 + 3*x - 1)*t^4 + ...
Q_n(x) can be defined by the recurrence relation Q_n(x) = (x + 1)*Q_(n-1)(x) + (x - 1)*Q_(n-2)(x), Q_0(x)=0, Q_1(x)=1.
Discriminants of Q_n(x) gives the sequence: 0, 1, 1, 9, 320, 35600, 10948608, 8664190976, 16836271800320, 77757312009240576, 833309554769920000000, 20346889104219547132493824,...
Q_n(0)  = A128834(n).
Q_n(1)  = A000079(n-1), n>0.
Q_n(2)  = A006190(n).
Q_n(3)  = A090017(n).
Q_n(4)  = A015536(n).
Q_n(5)  = A135032(n).
Q_n(6)  = A015562(n).
Q_n(7)  = A190560(n).
Q_n(8)  = A015583(n).
Q_n(9)  = A190957(n).
Q_n(10) = A015603(n).

			Triangle begins:
   1;
   1,  1;
   0,  3,  1;
  -1,  3,  5,  1;
  -1, -1, 10,  7,  1;
...
The first few polynomials are:
Q_0(x) = 0;
Q_1(x) = 1;
Q_2(x) = x + 1;
Q_3(x) = x^2 + 3*x;
Q_4(x) = x^3 + 5*x^2 + 3*x - 1;
Q_5(x) = x^4 + 7*x^3 + 10*x^2 - x - 1,
...

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Ilya Gutkovskiy, Polynomials Q_n(x)
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A049310.

Flatten[Table[CoefficientList[((x + Sqrt[x (x + 6) - 3] + 1)^n - (x - Sqrt[x (x + 6) - 3] + 1)^n)/2^n/Sqrt[x (x + 6) - 3], x], {n, 0, 13}]]

cp's OEIS Frontend

A254657 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2}.

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

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A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

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cp's OEIS Frontend

A254657 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2}.

Views

Author

Keywords

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Magma

Mathematica

PARI

Formula

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)*((x + sqrt(x*(x + 6) - 3) + 1)^n - (x - sqrt(x*(x + 6) - 3) + 1)^n)/sqrt(x*(x + 6) - 3).

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).