A190958 -id:A190958 - cp's OEIS frontend

A190968 a(n) = 4a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 6, -16, -124, -336, -104, 2944, 12816, 21824, -40864, -381696, -1118144, -655616, 8558976, 40792064, 77578496, -97606656, -1166211584, -3688779776, -3093003264, 24515784704, 128993171456, 270814838784, -206672359424, -3534837825536, -12072627707904

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

sign

Michael De Vlieger, Table of n, a(n) for n = 0..2000
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,-10).

Cf. A190958 (index to generalized Fibonacci sequences).

Mathematica
```
LinearRecurrence[{4,-10}, {0,1}, 50]
```

G.f.: x/(1 - 4*x + 10*x^2). - Philippe Deléham, Oct 12 2011

A190971 a(n) = 5a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 15, 25, -25, -375, -1625, -4375, -5625, 15625, 134375, 515625, 1234375, 1015625, -7265625, -46484375, -159765625, -333984375, -72265625, 2978515625, 15615234375, 48291015625, 85302734375, -56396484375, -1135009765625, -5111083984375, -14205322265625

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

sign

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

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 5*(Self(n-1) - 2*Self(n-2)): n in [1..51]]; // G. C. Greubel, Jun 10 2022

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

concat(0, Vec(x/(1-5*x+10*x^2) + O(x^100))) \\ Altug Alkan, Nov 26 2015

[lucas_number1(n,5,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

a(n) = (i/sqrt(15))*(((5 - i*sqrt(15))/2)^n - ((5 + i*sqrt(15))/2)^n). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 5*x + 10*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Jun 10 2022: (Start)
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, sqrt(10)/4).
E.g.f.: (2/sqrt(15))*exp(5*x/2)*sin(sqrt(15)*x/2). (End)

A190981 a(n) = 9a(n-1) - 4a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 77, 657, 5605, 47817, 407933, 3480129, 29689429, 253284345, 2160801389, 18434075121, 157263470533, 1341634934313, 11445660526685, 97644405002913, 833017002919477, 7106575406263641, 60627110644694861, 517217694177199185, 4412450805016013221

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,-4).

Cf. A190958 (index to generalized Fibonacci sequences).

[2^(n-1)*Evaluate(ChebyshevU(n), 9/4): n in [0..30]]; // G. C. Greubel, Aug 25 2022

Mathematica
```
LinearRecurrence[{9,-4}, {0,1}, 50]
```

A190981 = BinaryRecurrenceSequence(9,-4,0,1)
[A190981(n) for n in (0..30)] # G. C. Greubel, Aug 25 2022

G.f.: x/(1-9x+4*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (2/sqrt(65))*exp(9*x/2)*sinh(sqrt(65)*x/2). - G. C. Greubel, Aug 25 2022

A190982 a(n) = 9a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 76, 639, 5371, 45144, 379441, 3189249, 26806036, 225308079, 1893742531, 15917142384, 133785568801, 1124484407289, 9451431821596, 79440464357919, 667707020113291, 5612160859230024, 47170912632503761, 396477409396383729, 3332442121404934756

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 (9,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

[Round(5^((n-1)/2)*Evaluate(ChebyshevU(n), 9/(2*Sqrt(5)))): n in [0..30]]; // G. C. Greubel, Aug 26 2022

Mathematica
```
LinearRecurrence[{9,-5}, {0,1}, 50]
```

A190982 = BinaryRecurrenceSequence(9,-5,0,1)
[A190982(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

G.f.: x/(1 - 9*x + 5*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (2/sqrt(61))*exp(9*x/2)*sinh(sqrt(61)*x/2). - G. C. Greubel, Aug 26 2022

A190985 a(n) = 10a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 97, 940, 9109, 88270, 855373, 8288920, 80323081, 778364050, 7542671257, 73091620420, 708288190429, 6863607043030, 66511205859013, 644521237461040, 6245678757033361, 60523223857950490, 586495202308404817, 5683382351510196700, 55074337908176752549

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-3).

Cf. A190958 (index to generalized Fibonacci sequences).

[Round(3^((n-1)/2)*Evaluate(ChebyshevU(n), 5/Sqrt(3))): n in [0..30]]; // G. C. Greubel, Sep 03 2022

Mathematica
```
LinearRecurrence[{10,-3}, {0,1}, 50]
```

A190985 = BinaryRecurrenceSequence(10, -3, 0, 1)
[A190985(n) for n in (0..30)] # G. C. Greubel, Sep 03 2022

G.f.: x/(1-10x+3*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (1/sqrt(22))*exp(5*x)*sinh(sqrt(22)*x). - G. C. Greubel, Sep 03 2022

A190986 a(n) = 10a(n-1) - 4a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 96, 920, 8816, 84480, 809536, 7757440, 74336256, 712332800, 6825982976, 65410498560, 626801053696, 6006368542720, 57556481212416, 551539337953280, 5285167454683136, 50645517195018240, 485314502131449856, 4650562952534425600, 44564371516818456576

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,-4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A004254, A049310.

[2^(n-1)*Evaluate(ChebyshevU(n-1), 5/2): n in [0..30]]; // G. C. Greubel, Sep 03 2022

Mathematica
```
LinearRecurrence[{10,-4}, {0,1}, 50]
```

A190986 = BinaryRecurrenceSequence(10, -4, 0, 1)
[A190986(n) for n in (0..30)] # G. C. Greubel, Sep 03 2022

G.f.: x/(1 - 10*x + 4*x^2). - Philippe Deléham, Oct 12 2011
E.g.f.: (1/sqrt(21))*exp(5*x)*sinh(sqrt(21)*x). - G. C. Greubel, Sep 03 2022
a(n) = 2^(n-1)*S(n-1, 5), with the Chebyshev S-polynomial (see A049310) S(n-1, 5) = A004254(n). See the Magma program by G. C. Greubel. - Wolfdieter Lang, Nov 15 2023

cp's OEIS Frontend

A190968 a(n) = 4*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=1.

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Mathematica

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A190971 a(n) = 5*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=1.

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Magma

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A190981 a(n) = 9*a(n-1) - 4*a(n-2), with a(0)=0, a(1)=1.

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Author

Keywords

Links

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Programs

Magma

Mathematica

SageMath

Formula

A190982 a(n) = 9*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A190985 a(n) = 10*a(n-1) - 3*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A190986 a(n) = 10*a(n-1) - 4*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A190968 a(n) = 4a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.

A190971 a(n) = 5a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.

A190981 a(n) = 9a(n-1) - 4a(n-2), with a(0)=0, a(1)=1.

A190982 a(n) = 9a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A190985 a(n) = 10a(n-1) - 3a(n-2), with a(0)=0, a(1)=1.

A190986 a(n) = 10a(n-1) - 4a(n-2), with a(0)=0, a(1)=1.