A190958 -id:A190958 - cp's OEIS frontend

A190967 a(n) = 4a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 7, -8, -95, -308, -377, 1264, 8449, 22420, 13639, -147224, -711647, -1521572, 318535, 14968288, 57006337, 93310756, -139814009, -1399052840, -4337885279, -4760065556, 20000705287, 122843411152, 311367297025, 139878487732, -2242791722297

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Michael De Vlieger, Table of n, a(n) for n = 0..2096
Mei Bai, Wenchang Chu, and Dongwei Guo, Reciprocal Formulae among Pell and Lucas Polynomials, Mathematics, 10, 2691, (2022). See p. 5.
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-9).

Cf. A190958 (index to generalized Fibonacci sequences).

a:=[0,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Dec 07 2019

I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 07 2019

seq(coeff(series(x/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 07 2019

LinearRecurrence[{4,-9}, {0,1}, 50]
Table[FullSimplify[(3*I)^(n-1)*Fibonacci[n, -4*I/3]], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x/(1-4*x+9*x^2))) \\ G. C. Greubel, Dec 07 2019

def A190967_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/(1-4*x+9*x^2) ).list()
A190967_list(30) # G. C. Greubel, Dec 07 2019

G.f.: x/(1-4*x+9*x^2). - Philippe Deléham, Oct 12 2011
a(n) = (3*i)^(n-1)*Fibonacci(n, -4*i/3), where i=sqrt(-1) and F(n,x) is the Fibonacci polynomial. - G. C. Greubel, Dec 07 2019
a(n) = (3^n*sin(n*arccos(2/3)))/sqrt(5) = 3^(n-1)*chebyshevU(n-1, 2/3). - Federico Provvedi, Feb 23 2022
E.g.f.: exp(2*x)*sin(sqrt(5)*x)/sqrt(5). - Stefano Spezia, Jul 24 2025

A190975 a(n) = 8a(n-1) - 2a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 62, 480, 3716, 28768, 222712, 1724160, 13347856, 103334528, 799980512, 6193175040, 47945439296, 371177164288, 2873526435712, 22245857157120, 172219804385536, 1333266720770048, 10321694157389312, 79907019817574400, 618612770225816576

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6,7} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).
Index entries for linear recurrences with constant coefficients, signature (8, -2).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 8*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

Mathematica
```
LinearRecurrence[{8,-2}, {0,1}, 50]
```

Vec(1/(1-8*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

a(n) = ((4 + sqrt(14))^n - (4 - sqrt(14))^n)/(2*sqrt(14)). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 8x + 2*x^2). - Philippe Deléham, Oct 12 2011
E.g.f.: (1/sqrt(14))*exp(4*x)*sinh(sqrt(14)*x). - G. C. Greubel, Dec 18 2015

A190976 a(n) = 8a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 61, 464, 3529, 26840, 204133, 1552544, 11807953, 89805992, 683024077, 5194774640, 39509124889, 300488675192, 2285382026869, 17381590189376, 132196575434401, 1005427832907080, 7646832936953437, 58158379996906256, 442326541164389737

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-3).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 8*Self(n-1) - 3*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 11 2022

Mathematica
```
LinearRecurrence[{8,-3}, {0,1}, 50]
```

[lucas_number1(n,8,3) for n in (0..50)] # G. C. Greubel, Jun 11 2022

a(n) = ((4 + sqrt(13))^n - (4 - sqrt(13))^n)/(2*sqrt(13)). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 8*x + 3*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Jun 11 2022: (Start)
a(n) = 3^((n-1)/2)*ChebyshevU(n-1, 4/sqrt(3)).
E.g.f.: (1/sqrt(13))*exp(4*x)*sinh(sqrt(13)*x). (End)

A190979 a(n) = 9a(n-1) - 2a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 79, 693, 6079, 53325, 467767, 4103253, 35993743, 315737181, 2769647143, 24295349925, 213118855039, 1869478995501, 16399073249431, 143852701253877, 1261876164786031, 11069180080566525, 97098868395526663, 851751455398606917, 7471565361796408927

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6,7,8} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).
Index entries for linear recurrences with constant coefficients, signature (9,-2).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 10*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

Mathematica
```
LinearRecurrence[{9,-2}, {0,1}, 50]
```

concat(0, Vec(x/(1-9*x+2*x^2) + O(x^30))) \\ Colin Barker, Feb 26 2016

A190979 = BinaryRecurrenceSequence(9,-2,0,1)
[A190979(n) for n in (0..30)] # G. C. Greubel, Jun 17 2022

a(n) = ((9/2 + 1/2*sqrt(73))^n - (9/2 - 1/2*sqrt(73))^n)/sqrt(79). - Giorgio Balzarotti, May 27 2011
G.f.: x / (1-9*x+2*x^2). - Colin Barker, Feb 26 2016
From G. C. Greubel, Jun 17 2022: (Start)
a(n) = 2^((n-1)/2)*ChebyshevU(n-1, 9*x/(2*sqrt(2))).
E.g.f.: (2/sqrt(73))*exp(9*x/2)*sinh(sqrt(73)*x/2). (End)

A190980 a(n) = 9a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 78, 675, 5841, 50544, 437373, 3784725, 32750406, 283399479, 2452344093, 21220898400, 183631053321, 1589016784689, 13750257902238, 118985270766075, 1029616663187961, 8909594156393424, 77097497417976933, 667148694292612125, 5773045756379578326

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,-3).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

LinearRecurrence[{9,-3}, {0,1}, 50]
CoefficientList[Series[x/(1-9*x+3*x^2), {x,0,50}], x] (* G. C. Greubel, Jan 14 2018 *)

x='x+O('x^30); concat([0], Vec(x/(1-9*x+3*x^2))) \\ G. C. Greubel, Jan 14 2018

G.f.: x/(1-9*x+3*x^2). - Philippe Deléham, Oct 12 2011

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 95, 900, 8525, 80750, 764875, 7245000, 68625625, 650031250, 6157184375, 58321687500, 552430953125, 5232701093750, 49564856171875, 469485056250000, 4447026281640625, 42122837535156250, 398993243943359375, 3779318251757812500, 35798216297861328125

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A019934 (sqrt(5-2*sqrt(5))), A019952 (sqrt(5+2*sqrt(5))).

[Round(5^((n-1)/2)*Evaluate(ChebyshevU(n), Sqrt(5))): n in [0..30]]; // G. C. Greubel, Sep 07 2022

Mathematica
```
LinearRecurrence[{10,-5}, {0,1}, 50]
```

A190987 = BinaryRecurrenceSequence(10, -5, 0, 1)
[A190987(n) for n in (0..30)] # G. C. Greubel, Sep 07 2022

G.f.: x/(1 - 10*x + 5*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (1/(2*sqrt(5)))*exp(5*x)*sinh(2*sqrt(5)*x). - G. C. Greubel, Sep 07 2022

A190989 a(n) = 10a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 93, 860, 7949, 73470, 679057, 6276280, 58009401, 536160050, 4955534693, 45802226580, 423333522949, 3912719643430, 36163861773657, 334249580232560, 3089348769910001, 28553740637472090, 263911964985350893, 2439243465391204300, 22545050899014586749

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-7).

Cf. A190958 (index to generalized Fibonacci sequences)

[Round(7^((n-1)/2)*Evaluate(ChebyshevU(n), 5/Sqrt(7))): n in [0..30]]; // G. C. Greubel, Sep 15 2022

Mathematica
```
LinearRecurrence[{10,-7}, {0,1}, 50]
```

A190989 = BinaryRecurrenceSequence(10, -7, 0, 1)
[A190989(n) for n in (0..30)] # G. C. Greubel, Sep 15 2022

G.f.: x/ ( 1-10*x+7*x^2 ). - R. J. Mathar, May 26 2011

E.g.f.: (1/(3*sqrt(2)))*exp(5*x)*sinh(3*sqrt(2)*x). - G. C. Greubel, Sep 16 2022

A120743 a(n) = (1/2)(1 + 3i)^n + (1/2)(1 - 3i)^n where i = sqrt(-1).

Original entry on oeis.org

1, -8, -26, 28, 316, 352, -2456, -8432, 7696, 99712, 122464, -752192, -2729024, 2063872, 31417984, 42197248, -229785344, -881543168, 534767104, 9884965888, 14422260736, -70005137408, -284232882176, 131585609728, 3105500041216

Offset: 1

Creighton Dement, Jun 11 2007

From R. J. Mathar, Jun 15 2007: (Start)
These are the row sums of the triangle A013610 after every 2nd column is deleted, then every 2nd column reversed in sign, creating an intermediate irregular triangle with entries C(n,2*k)*(-9)^k, k = 0..floor(n/2):
  1;
  1,   -9;
  1,  -27;
  1,  -54,    81;
  1,  -90,   405;
  1, -135,  1215,    -729;
  1, -189,  2835,   -5103;
  1, -252,  5670,  -20412,   6561;
  1, -324, 10206,  -61236,  59049;
  1, -405, 17010, -153090, 295245, -59049; (End)
Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[A*B] with A = + 1.5i' + .5j' + .5k' + .5e and B = 'ji' + e

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (2, -10).

Cf. A006495

[ n eq 1 select 1 else n eq 2 select -8 else 2*Self(n-1) -10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 24 2011

LinearRecurrence[{2,-10}, {1,-8}, 30] (* G. C. Greubel, Nov 09 2018 *)

x='x+O('x^30); Vec((1-10*x)/(1-2*x+10*x^2)) \\ G. C. Greubel, Nov 09 2018

a(n) = 2*a(n-1) - 10*a(n-2).
G.f.: x*(1-10*x)/(10*x^2 - 2*x + 1).
a(n) mod 9 = 1. - Paul Curtz, Apr 20 2011
G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(9*k+1)/(x*(9*k+10) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
E.g.f.: exp(x)*cos(3*x). - Sergei N. Gladkovskii, May 29 2013
a(n) = A190958(n)-10*A190958(n-1). - R. J. Mathar, Dec 13 2022

a(0)=1: a(n) is main diagonal of A009116(n). - Paul Curtz, Jul 22 2011

Edited by Jon E. Schoenfield, Nov 09 2018

A176333 Expansion of (1-3x)/(1-4x+9*x^2).

Original entry on oeis.org

1, 1, -5, -29, -71, -23, 547, 2395, 4657, -2927, -53621, -188141, -269975, 613369, 4883251, 14012683, 12101473, -77708255, -419746277, -979610813, -140726759, 8253590281, 34280901955, 62841295291, -57162936431, -794223403343, -2662427185493, -3501698111885

Offset: 0

Paul Barry, Apr 15 2010

Hankel transform of A176332.

Michael De Vlieger, Table of n, a(n) for n = 0..2097
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-9).

Cf. A176332, A190958, A190967.

a:=[1,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Dec 07 2019

I:=[1,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 07 2019

seq(coeff(series((1-3*x)/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 07 2019

CoefficientList[Series[(1-3x)/(1-4x+9x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,-9},{1,1},30] (* Harvey P. Dale, Sep 17 2012 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-4*x+9*x^2)) \\ G. C. Greubel, Dec 07 2019

def A176333_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-4*x+9*x^2) ).list()
A176333_list(30) # G. C. Greubel, Dec 07 2019

a(n) = 3^n*( cos(2*n*atan(1/sqrt(5))) - sin(2*n*atan(1/sqrt(5)))/sqrt(5) ).
a(0)=1, a(1)=1, a(n) = 4*a(n-1) - 9*a(n-2). - Harvey P. Dale, Sep 17 2012
a(n) = -3*A190967(n) + A190967(n+1). - R. J. Mathar, May 04 2013

A190966 a(n) = 4a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 8, 0, -64, -256, -512, 0, 4096, 16384, 32768, 0, -262144, -1048576, -2097152, 0, 16777216, 67108864, 134217728, 0, -1073741824, -4294967296, -8589934592, 0, 68719476736, 274877906944, 549755813888, 0, -4398046511104, -17592186044416, -35184372088832

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-8).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 4*(Self(n-1) -2*Self(n-2)): n in [1..41]]; // G. C. Greubel, Jan 10 2024

Mathematica
```
LinearRecurrence[{4,-8}, {0,1}, 50]
```

A190966=BinaryRecurrenceSequence(4,-8,0,1)
[A190966(n) for n in range(41)] # G. C. Greubel, Jan 10 2024

G.f.: x/(1 - 4*x + 8*x^2). - Philippe Deléham, Oct 12 2011
a(n) = 2^(n-1)*A009545(n). - R. J. Mathar, Apr 07 2022
From G. C. Greubel, Jan 10 2024: (Start)
a(n) = 8^((n-1)/2)*ChebyshevU(n-1, 1/sqrt(2)).
E.g.f.: (1/2)*exp(2*x)*sin(2*x). (End)
a(n) = (i/4)*((2 - 2*i)^n - (2 + 2*i)^n), where i=sqrt(-1). - Taras Goy, Jan 04 2025

cp's OEIS Frontend

A190967 a(n) = 4*a(n-1) - 9*a(n-2), with a(0)=0, a(1)=1.

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A190975 a(n) = 8*a(n-1) - 2*a(n-2), with a(0)=0, a(1)=1.

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A190976 a(n) = 8*a(n-1) - 3*a(n-2), with a(0)=0, a(1)=1.

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A190979 a(n) = 9*a(n-1) - 2*a(n-2), with a(0)=0, a(1)=1.

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Magma

Mathematica

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A190980 a(n) = 9*a(n-1) - 3*a(n-2), with a(0)=0, a(1)=1.

Views

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Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A190987 a(n) = 10*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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Links

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Programs

Magma

Mathematica

SageMath

Formula

A190989 a(n) = 10*a(n-1) - 7*a(n-2), with a(0)=0, a(1)=1.

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A190967 a(n) = 4a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

A190975 a(n) = 8a(n-1) - 2a(n-2), with a(0)=0, a(1)=1.

A190976 a(n) = 8a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

A190979 a(n) = 9a(n-1) - 2a(n-2), with a(0)=0, a(1)=1.

A190980 a(n) = 9a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A190989 a(n) = 10a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

A120743 a(n) = (1/2)(1 + 3i)^n + (1/2)(1 - 3i)^n where i = sqrt(-1).

A176333 Expansion of (1-3x)/(1-4x+9*x^2).

A190966 a(n) = 4a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.