A190958 -id:A190958 - cp's OEIS frontend

A190969 a(n) = 5a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

0, 1, 5, 17, 45, 89, 85, -287, -2115, -8279, -24475, -56143, -84915, 24569, 802165, 3814273, 12654045, 32756041, 62547845, 50690897, -246928275, -1640168551, -6225416555, -18005734367, -40225339395, -57080822039, 36398604965, 638639601137, 2902009165965

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Let S(p):=Sum_{k=0..p-1} a(4k)*binomial(2k,k)^3/(-4096)^k. Zhi-Wei Sun conjectured that S(p) == 0 (mod p^2) for every odd prime p, and also S(p) == 0 (mod p^3) for any odd prime p == 1,2,4 (mod 7). - Zhi-Wei Sun, Mar 13 2013

(a(n) + ((-1)^n)*n) mod 7 = 0 for n > 0; division yields following signed integer sequence: {0, 1, 2, 7, 12, 13, -42, -301, -1184, -3495, -8022, -12129, 3508, 114597, ...} with g.f.: (x - x^2)/((1 + x)^2 * (1 - 5*x + 8*x^2)). - Alexander R. Povolotsky, Mar 13 2013

Zhi-Wei Sun, Table of n, a(n) for n = 0..100
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
Index entries for linear recurrences with constant coefficients, signature (5,-8).

Cf. A190958 (index to generalized Fibonacci sequences).

Mathematica
```
LinearRecurrence[{5,-8}, {0,1}, 50]
```

a[0]:0$ a[1]:1$ a[n]:=5*a[n-1] - 8*a[n-2]$ makelist(a[n], n,0, 50); /* Martin Ettl, Oct 21 2012 */

G.f.: x/(1-5x+8*x^2). - Philippe Deléham, Oct 12 2011

A190973 a(n) = 7a(n-1) - 4a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 7, 45, 287, 1829, 11655, 74269, 473263, 3015765, 19217303, 122458061, 780337215, 4972528261, 31686348967, 201914329725, 1286654912207, 8198927066549, 52245869817015, 332925380452909, 2121494183902303, 13518757765504485, 86145327622922183

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 (7,-4).

Cf. A190958 (index to generalized Fibonacci sequences).

a:=[0,1];; for n in [3..30] do a[n]:=7*a[n-1]-4*a[n-2]; od; a; # G. C. Greubel, Dec 24 2019

I:=[0,1]; [n le 2 select I[n] else 7*Self(n-1) - 4*Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 24 2019

seq( simplify(2^(n-1)*ChebyshevU(n-1, 7/4)), n=0..30); # G. C. Greubel, Dec 24 2019

LinearRecurrence[{7, -4}, {0, 1}, 30]
Table[2^(n - 1) * ChebyshevU[n - 1, 7/4], {n, 0, 30}] (* G. C. Greubel, Dec 24 2019 *)

vector(31, n, 2^(n-2)*polchebyshev(n-2, 2, 7/4) ) \\ G. C. Greubel, Dec 24 2019

[2^(n-1)*chebyshev_U(n-1,7/4) for n in (0..30)] # G. C. Greubel, Dec 24 2019

a(n) = ((7/2 + 1/2*sqrt(33))^n - (7/2 - 1/2*sqrt(33))^n)/sqrt(33). - Giorgio Balzarotti, May 28 2011

G.f.: x/(1 - 7*x + 4*x^2). - Philippe Deléham, Oct 12 2011

A190977 a(n) = 8a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 59, 432, 3161, 23128, 169219, 1238112, 9058801, 66279848, 484944779, 3548158992, 25960548041, 189943589368, 1389745974739, 10168249851072, 74397268934881, 544336902223688, 3982708873115099, 29139986473802352, 213206347424843321, 1559950847029734808

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 (8,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 8*Self(n-1) -5*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 17 2022

Mathematica
```
LinearRecurrence[{8,-5}, {0,1}, 50]
```

[sum( (-1)^k*binomial(n-k-1, k)*5^k*8^(n-2*k-1) for k in (0..((n-1)//2))) for n in (0..40)] # G. C. Greubel, Jun 17 2022

a(n) = ((4 + sqrt(11))^n - (4 - sqrt(11))^n)/(2*sqrt(11)). - Giorgio Balzarotti, May 28 2011
G.f.: x/(1 - 8*x + 5*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Jun 17 2022: (Start)
a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 4/sqrt(5)).
E.g.f.: (1/sqrt(11))*exp(4*x)*sinh(sqrt(11)*x). (End)

A190983 a(n) = 9a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 9, 75, 621, 5139, 42525, 351891, 2911869, 24095475, 199388061, 1649919699, 13652948925, 112977022131, 934875505629, 7736017417875, 64014903727101, 529718029036659, 4383372838967325, 36272047376485971, 300148189354569789, 2483701419932212275

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,-6).

Cf. A190958 (index to generalized Fibonacci sequences).

[n le 2 select n-1 else 9*Self(n-1) - 6*Self(n-2):n in [1..22]]; // Marius A. Burtea, Jan 22 2020

R:=PowerSeriesRing(Integers(), 22); [0] cat Coefficients(R!( x/(1-9*x+6*x^2))); // Marius A. Burtea, Jan 22 2020

LinearRecurrence[{9,-6}, {0,1}, 50]
With[{s=Sqrt[57]},Table[Simplify[(2^(-1-x) (4s (9+s)^x-(9-s)^x (171+ 23s)))/ (57(9+s))],{x,30}]] (* Harvey P. Dale, Sep 01 2014 *)

A190983 = BinaryRecurrenceSequence(9,-6,0,1)
[A190983(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

G.f.: x/(1-9*x+6*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (2/sqrt(57))*exp(9*x/2)*sinh(sqrt(57)*x/2). - G. C. Greubel, Aug 26 2022

A190988 a(n) = 10a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 94, 880, 8236, 77080, 721384, 6751360, 63185296, 591344800, 5534336224, 51795293440, 484746917056, 4536697409920, 42458492596864, 397364741509120, 3718896459510016, 34804776146045440, 325734382703394304, 3048515170157670400, 28530745405356338176

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,-6).

Cf. A190958 (index to generalized Fibonacci sequences).

[Round(6^((n-1)/2)*Evaluate(ChebyshevU(n), 5/Sqrt(6))): n in [0..30]]; // G. C. Greubel, Sep 15 2022

Mathematica
```
LinearRecurrence[{10,-6}, {0,1}, 50]
```

A190988 = BinaryRecurrenceSequence(10, -6, 0, 1)
[A190988(n) for n in (0..30)] # G. C. Greubel, Sep 15 2022

G.f.: x/(1-10*x+6*x^2). - Philippe Deléham, Oct 12 2011
From G. C. Greubel, Sep 15 2022: (Start)
a(n) = 6^((n-1)/2) * ChebyshevU(n-1, 5/sqrt(6)).
E.g.f.: (1/sqrt(19))*exp(5*x)*sinh(sqrt(19)*x). (End)

A190959 a(n) = 3a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 4, -3, -29, -72, -71, 147, 796, 1653, 979, -5328, -20879, -35997, -3596, 169197, 525571, 730728, -435671, -4960653, -12703604, -13307547, 23595379, 137323872, 293994721, 195364803, -883879196, -3628461603, -6465988829, -1255658472, 28562968729

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

This is the Lucas U(P=3, Q=5) sequence. - R. J. Mathar, Oct 24 2012

a(n+2)/a(n+1) equals the continued fraction 3 - 5/(3 - 5/(3 - 5/(3 - ... - 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (3,-5).

Cf. A190958 (index to generalized Fibonacci sequences), A190970 (binomial transf.), A106852 (inv. bin. transf., shifted).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica
```
LinearRecurrence[{3,-5}, {0,1}, 50]
```

x='x+O('x^30); concat([0], Vec(x/(1-3x+5*x^2))) \\ G. C. Greubel, Jan 25 2018

G.f.: x/(1 - 3*x + 5*x^2). - Philippe Deléham, Oct 11 2011

E.g.f.: 2*exp(3*x/2)*sin(sqrt(11)*x/2)/sqrt(11). - Stefano Spezia, Oct 06 2019

A190961 a(n) = 3a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 2, -15, -59, -72, 197, 1095, 1906, -1947, -19183, -43920, 2521, 315003, 927362, 577065, -4760339, -18320472, -21639043, 63326175, 341451826, 581072253, -646946023, -6008343840, -13496409359, 1569178803, 99182401922, 286562954145, 165412048981

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 (3,-7).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica
```
LinearRecurrence[{3,-7}, {0,1}, 50]
```

x='x+O('x^30); concat([0], Vec(x/(1-3*x+7*x^2))) \\ G. C. Greubel, Jan 25 2018

G.f.: x/(1-3*x+7*x^2). - Philippe Deléham, Oct 11 2011

E.g.f.: 2*exp(3*x/2)*sin(sqrt(19)*x/2)/sqrt(19). - Stefano Spezia, Jul 28 2025

A190962 a(n) = 3a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 1, -21, -71, -45, 433, 1659, 1513, -8733, -38303, -45045, 171289, 874227, 1252369, -3236709, -19729079, -33293565, 57951937, 440204331, 856997497, -950642157, -9707906447, -21518582085, 13107505321, 211471172643, 529553475361, -103108955061

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 (3,-8).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

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

x='x+O('x^30); concat([0], Vec(x/(1-3*x+8*x^2))) \\ G. C. Greubel, Jan 25 2018

G.f.: x/(1-3*x+8*x^2). - Philippe Deléham, Oct 11 2011

A190963 a(n) = 3a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 0, -27, -81, 0, 729, 2187, 0, -19683, -59049, 0, 531441, 1594323, 0, -14348907, -43046721, 0, 387420489, 1162261467, 0, -10460353203, -31381059609, 0, 282429536481, 847288609443, 0, -7625597484987, -22876792454961, 0, 205891132094649

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 (3,-9).

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A128834.

[n le 2 select n-1 else 3*Self(n-1)-9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica
```
LinearRecurrence[{3,-9}, {0,1}, 50]
```

my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+9*x^2))) \\ G. C. Greubel, Jan 25 2018

A190963=BinaryRecurrenceSequence(3,-9,0,1)
[A190963(n) for n in range(41)] # G. C. Greubel, Jan 11 2024

G.f.: x/(1-3*x+9*x^2). - Philippe Deléham, Oct 11 2011
From G. C. Greubel, Jan 11 2024: (Start)
a(n) = 3^(n-1)*ChebyshevU(n-1, 1/2).
a(n) = 3^(n-1)*A128834(n).
E.g.f.: (2/(3*sqrt(3)))*exp(3*x/2)*sin(3*sqrt(3)*x/2). (End)

A190964 a(n) = 3a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, -1, -33, -89, 63, 1079, 2607, -2969, -34977, -75241, 124047, 1124551, 2133183, -4845961, -35869713, -59149529, 181248543, 1135240919, 1593237327, -6572697209, -35650464897, -41224422601, 232831381167, 1110738369511, 1003901296863, -8095679804521

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 (3,-10).

Cf. A190958 (index to generalized Fibonacci sequences).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

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

my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+10*x^2))) \\ G. C. Greubel, Jan 25 2018

A190964=BinaryRecurrenceSequence(3,-10,0,1)
[A190964(n) for n in range(41)] # G. C. Greubel, Jan 11 2024

G.f.: x/(1-3*x+10*x^2). - Philippe Deléham, Oct 11 2011
From G. C. Greubel, Jan 11 2024: (Start)
a(n) = 10^((n-1)/2)*ChebyshevU(n-1, 3/(2*sqrt(10))).
E.g.f.: (2/sqrt(31))*exp(3*x/2)*sin(sqrt(31)*x/2). (End)

cp's OEIS Frontend

A190969 a(n) = 5*a(n-1) - 8*a(n-2), with a(0)=0, a(1)=1.

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A190973 a(n) = 7*a(n-1) - 4*a(n-2), with a(0) = 0, a(1) = 1.

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A190977 a(n) = 8*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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A190983 a(n) = 9*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

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A190988 a(n) = 10*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

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A190959 a(n) = 3*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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Magma

Mathematica

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A190961 a(n) = 3*a(n-1) - 7*a(n-2), with a(0)=0, a(1)=1.

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A190969 a(n) = 5a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

A190973 a(n) = 7a(n-1) - 4a(n-2), with a(0) = 0, a(1) = 1.

A190977 a(n) = 8a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A190983 a(n) = 9a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

A190988 a(n) = 10a(n-1) - 6a(n-2), with a(0)=0, a(1)=1.

A190959 a(n) = 3a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A190961 a(n) = 3a(n-1) - 7a(n-2), with a(0)=0, a(1)=1.

A190962 a(n) = 3a(n-1) - 8a(n-2), with a(0)=0, a(1)=1.

A190963 a(n) = 3a(n-1) - 9a(n-2), with a(0)=0, a(1)=1.

A190964 a(n) = 3a(n-1) - 10a(n-2), with a(0)=0, a(1)=1.