A080040 -id:A080040 - cp's OEIS frontend

A189741 a(1)=4, a(2)=2, a(n) = 4a(n-1) + 2a(n-2).

4, 2, 16, 68, 304, 1352, 6016, 26768, 119104, 529952, 2358016, 10491968, 46683904, 207719552, 924246016, 4112423168, 18298184704, 81417585152, 362266710016, 1611902010368, 7172141461504, 31912369866752, 141993762390016, 631799789293568, 2811186681954304

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{4,2},{4,2},40]
```

a[1]:4$ a[2]:2$ a[n]:=4*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 25);  /* Bruno Berselli, May 24 2011 */

G.f.: 2*x*(2-7*x)/(1-4*x-2*x^2). - Bruno Berselli, May 24 2011

A189734 a(1)=2, a(2)=5, a(n)=2a(n-1) + 5a(n-2).

Original entry on oeis.org

2, 5, 20, 65, 230, 785, 2720, 9365, 32330, 111485, 384620, 1326665, 4576430, 15786185, 54454520, 187839965, 647952530, 2235104885, 7709972420, 26595469265, 91740800630, 316458947585, 1091621898320, 3765538534565, 12989186560730, 44806065794285

Offset: 1

Harvey P. Dale, Apr 26 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (2,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{2,5},{2,5},40]
```

a[1]:2$ a[2]:5$ a[n]:=2*a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

G.f.: x*(2+x)/(1-2*x-5*x^2). - Bruno Berselli, May 24 2011

A189736 a(1)=3, a(2)=2, a(n)=3a(n-1) + 2a(n-2).

Original entry on oeis.org

3, 2, 12, 40, 144, 512, 1824, 6496, 23136, 82400, 293472, 1045216, 3722592, 13258208, 47219808, 168175840, 598967136, 2133253088, 7597693536, 27059586784, 96374147424, 343241615840, 1222473142368, 4353902658784, 15506654261088, 55227768100832

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (3,2).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{3,2},{3,2},40]
```

a[1]:3$ a[2]:2$ a[n]:=3*a[n-1]+2*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

G.f.: x*(3-7*x)/(1-3*x-2*x^2). - Bruno Berselli, May 24 2011

A189742 a(1)=4, a(2)=3, a(n) = 4a(n-1) + 3a(n-2).

Original entry on oeis.org

4, 3, 24, 105, 492, 2283, 10608, 49281, 228948, 1063635, 4941384, 22956441, 106649916, 495468987, 2301825696, 10693709745, 49680316068, 230802393507, 1072250522232, 4981409269449, 23142388644492, 107513782386315, 499482295478736, 2320470529073889

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (4,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{4,3},{4,3},40]
```

a[1]:4$ a[2]:3$ a[n]:=4*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: x*(4-13*x)/(1-4*x-3*x^2). - Bruno Berselli, May 24 2011

A189743 a(1)=4, a(2)=4, a(n) = 4a(n-1) + 4a(n-2).

Original entry on oeis.org

4, 4, 32, 144, 704, 3392, 16384, 79104, 381952, 1844224, 8904704, 42995712, 207601664, 1002389504, 4839964672, 23369416704, 112837525504, 544827768832, 2630661177344, 12701955784704, 61330467848192, 296129694531584, 1429840649519104, 6903881376202752

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{4,4},{4,4},40]
```

a[1]:4$ a[2]:4$ a[n]:=4*a[n-1]+4*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

G.f.: 4*x*(1-3*x)/(1-4*x-4*x^2). - Bruno Berselli, May 24 2011

A189744 a(1)=4, a(2)=5, a(n) = 4a(n-1) + 5a(n-2).

Original entry on oeis.org

4, 5, 40, 185, 940, 4685, 23440, 117185, 585940, 2929685, 14648440, 73242185, 366210940, 1831054685, 9155273440, 45776367185, 228881835940, 1144409179685, 5722045898440, 28610229492185, 143051147460940, 715255737304685, 3576278686523440, 17881393432617185

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734-A189739, A189741, A189742, A189743, A189745-A189749.

[5/2*(-1)^(n-1)+3/2*5^(n-1): n in [1..30]]; // Vincenzo Librandi, Jul 15 2011

Mathematica
```
LinearRecurrence[{4,5},{4,5},40]
```

a[1]:4$ a[2]:5$ a[n]:=4*a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 24); /* Bruno Berselli, May 24 2011 */

a(n)=5/2*(-1)^(n-1)+3/2*5^(n-1) \\ Charles R Greathouse IV, Jul 02 2013

G.f.: x*(4-11*x)/(1 - 4*x - 5*x^2). - Bruno Berselli, May 24 2011

A189745 a(n) = 5*a(n-1) + a(n-2); with a(1)=5, a(2)=1.

Original entry on oeis.org

5, 1, 10, 51, 265, 1376, 7145, 37101, 192650, 1000351, 5194405, 26972376, 140056285, 727253801, 3776325290, 19608880251, 101820726545, 528712512976, 2745383291425, 14255628970101, 74023528141930, 384373269679751, 1995889876540685, 10363822652383176

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,1).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{5,1},{5,1},40]
```

a[1]:5$ a[2]:1$ a[n]:=5*a[n-1]+a[n-2]$ makelist(a[n], n, 1, 24); /*
 Bruno Berselli, May 24 2011 */

G.f.: x*(5-24*x)/(1-5*x-x^2). - Bruno Berselli, May 24 2011

A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

Original entry on oeis.org

5, 5, 50, 275, 1625, 9500, 55625, 325625, 1906250, 11159375, 65328125, 382437500, 2238828125, 13106328125, 76725781250, 449160546875, 2629431640625, 15392960937500, 90111962890625, 527524619140625, 3088182910156250, 18078537646484375, 105833602783203125

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (5,5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748.

Mathematica
```
LinearRecurrence[{5,5},{5,5},40]
```

a[1]:5$ a[2]:5$ a[n]:=5*a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */

G.f.: 5*x*(1-4*x)/(1-5*x-5*x^2). - Bruno Berselli, May 24 2011

a(n) = 5*A188168(n). - R. J. Mathar, Feb 13 2020

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 9, 36, 135, 513, 1944, 7371, 27945, 105948, 401679, 1522881, 5773680, 21889683, 82990089, 314639316, 1192888215, 4522582593, 17146412424, 65006985051, 246460192425, 934401532428, 3542585174559, 13430960120961

Offset: 0

Roger L. Bagula, May 29 2005

The first entry of the vector v[n] = M*v[n-1], where M is the 2 x 2 matrix [[0,3],[1,3]] and v[1] is the column vector [0,1]. The characteristic polynomial of the matrix M is x^2-3x-3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).

Equals 3*A030195(n).
Cf. A028860.
Cf. A002605, A026150, A028859, A080040, A083337, A108898, A125145.

a106435 n = a106435_list !! n
a106435_list = 0 : 3 : map (* 3) (zipWith (+) a106435_list (tail
a106435_list))
-- Reinhard Zumkeller, Oct 15 2011

a:=[0,3]; [n le 2 select a[n] else    3*Self(n-1) + 3*Self(n-2) : n in [1..24]]; // Marius A. Burtea, Jan 21 2020

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!(3*x/(1-3*x-3*x^2))); // Marius A. Burtea, Jan 21 2020

seq(coeff(series(3*x/(1-3*x-3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Mar 12 2020

LinearRecurrence[{3,3}, {0,3}, 30] (* G. C. Greubel, Mar 12 2020 *)

PARI
```
a(n)=([0,3;1,3]^n)[1,2]
```

[3^((n+1)/2)*i^(1-n)*chebyshev_U(n-1, i*sqrt(3)/2) for n in (0..30)] # G. C. Greubel, Mar 12 2020

G.f.: 3*x/(1-3*x-3*x^2). - Philippe Deléham, Nov 19 2008
From G. C. Greubel, Mar 12 2020: (Start)
a(n) = 3^((n+1)/2) * Fibonacci(n, sqrt(3)), where F(n, x) is the Fibonacci polynomial.
a(n) = 3^((n+1)/2)*i^(1-n)*ChebyshevU(n-1, i*sqrt(3)/2). (End)

Edited by N. J. A. Sloane, May 20 2006 and May 29 2006

Offset corrected by Reinhard Zumkeller, Oct 15 2011

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).

Original entry on oeis.org

3, 3, 18, 63, 243, 918, 3483, 13203, 50058, 189783, 719523, 2727918, 10342323, 39210723, 148659138, 563609583, 2136806163, 8101247238, 30714160203, 116446222323, 441481147578, 1673782109703, 6345789771843, 24058715644638, 91213516249443, 345816695682243

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189732, A189734, A189735, A189736, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{3,3},{3,3},40]
```

a[1]:3$ a[2]:3$ a[n]:=3*a[n-1]+3*a[n-2]$ makelist(a[n], n, 1, 26); /* Bruno Berselli, May 24 2011 */

G.f.: 3*x*(1-2*x)/(1-3*x-3*x^2). - Bruno Berselli, May 24 2011

cp's OEIS Frontend

A189741 a(1)=4, a(2)=2, a(n) = 4*a(n-1) + 2*a(n-2).

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A189734 a(1)=2, a(2)=5, a(n)=2*a(n-1) + 5*a(n-2).

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A189736 a(1)=3, a(2)=2, a(n)=3*a(n-1) + 2*a(n-2).

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Maxima

Formula

A189742 a(1)=4, a(2)=3, a(n) = 4*a(n-1) + 3*a(n-2).

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Maxima

Formula

A189743 a(1)=4, a(2)=4, a(n) = 4*a(n-1) + 4*a(n-2).

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Maxima

Formula

A189744 a(1)=4, a(2)=5, a(n) = 4*a(n-1) + 5*a(n-2).

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Magma

Mathematica

Maxima

PARI

Formula

A189745 a(n) = 5*a(n-1) + a(n-2); with a(1)=5, a(2)=1.

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Links

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Programs

Mathematica

Maxima

Formula

A189749 a(1)=5, a(2)=5, a(n)=5*a(n-1) + 5*a(n-2).

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Programs

A189741 a(1)=4, a(2)=2, a(n) = 4a(n-1) + 2a(n-2).

A189734 a(1)=2, a(2)=5, a(n)=2a(n-1) + 5a(n-2).

A189736 a(1)=3, a(2)=2, a(n)=3a(n-1) + 2a(n-2).

A189742 a(1)=4, a(2)=3, a(n) = 4a(n-1) + 3a(n-2).

A189743 a(1)=4, a(2)=4, a(n) = 4a(n-1) + 4a(n-2).

A189744 a(1)=4, a(2)=5, a(n) = 4a(n-1) + 5a(n-2).

A189749 a(1)=5, a(2)=5, a(n)=5a(n-1) + 5a(n-2).

A106435 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=3.

A189737 a(1)=3, a(2)=3, a(n)=3a(n-1) + 3a(n-2).