A107585 -id:A107585 - cp's OEIS frontend

A107583 a(n) = 3^n - 3*n.

1, 0, 3, 18, 69, 228, 711, 2166, 6537, 19656, 59019, 177114, 531405, 1594284, 4782927, 14348862, 43046673, 129140112, 387420435, 1162261410, 3486784341, 10460353140, 31381059543, 94143178758, 282429536409, 847288609368, 2541865828251, 7625597484906, 22876792454877

Offset: 0

Zak Seidov, May 16 2005

a(n) is the number k such that the number m with n 3's and k 1's has digit product = digit sum = 3^n.

			Corresponding numbers m are 1, 3, 11133, 111111111111111111333, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A107584, A107585.

[3^n-3*n: n in [0..30]]; // Vincenzo Librandi, Oct 22 2011

Mathematica
```
Table[3^m-3*m, {m, 0, 20}]
```

a(n)=3^n-3*n \\ Charles R Greathouse IV, Sep 08 2012

From Elmo R. Oliveira, Sep 09 2024: (Start)
G.f.: (1 - 5*x + 10*x^2)/((1 - 3*x)*(1 - x)^2).
E.g.f.: exp(x)*(exp(2*x) - 3*x).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3) for n > 2. (End)

Corrected by Charles R Greathouse IV, Sep 08 2012

A107584 a(n) = 4^n - 4*n.

Original entry on oeis.org

1, 0, 8, 52, 240, 1004, 4072, 16356, 65504, 262108, 1048536, 4194260, 16777168, 67108812, 268435400, 1073741764, 4294967232, 17179869116, 68719476664, 274877906868, 1099511627696, 4398046511020, 17592186044328, 70368744177572, 281474976710560, 1125899906842524

Offset: 0

Zak Seidov, May 16 2005

Numbers a(n) = k such that number m with n 4's and k 1's has digit product = digit sum = 4^n.

			Corresponding numbers m are 1, 4, 1111111144, ...

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A107583, A107585.

[(4^n - 4*n): n in [0..25]]; // Vincenzo Librandi, Dec 16 2010

Table[4^m-4*m, {m, 0, 20}]
LinearRecurrence[{6,-9,4},{1,0,8},30] (* Harvey P. Dale, Oct 21 2011 *)

a(n)=4^n-4*n \\ Charles R Greathouse IV, Sep 08 2012

def A107584(n): return (1<<(n<<1))-(n<<2) # Chai Wah Wu, Nov 29 2023

From Harvey P. Dale, Oct 21 2011: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=0, and a(2)=8.
G.f.: (-17*x^2+6*x-1)/((x-1)^2*(4*x-1)). (End)
E.g.f.: exp(x)*(exp(3*x) - 4*x). - Elmo R. Oliveira, Sep 10 2024

More terms from Vincenzo Librandi, Dec 16 2010

Corrected by Charles R Greathouse IV, Sep 08 2012

A198396 a(n) = 6^n - 6*n.

Original entry on oeis.org

1, 0, 24, 198, 1272, 7746, 46620, 279894, 1679568, 10077642, 60466116, 362796990, 2176782264, 13060693938, 78364164012, 470184984486, 2821109907360, 16926659444634, 101559956668308, 609359740010382, 3656158440062856, 21936950640377730, 131621703842267004, 789730223053602678

Offset: 0

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[6^n-6*n: n in [0..25]];
```

CoefficientList[Series[(1 - 8*x + 37*x^2)/((1 - 6*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
Table[6^n-6n,{n,0,30}] (* or *) LinearRecurrence[{8,-13,6},{1,0,24},30] (* Harvey P. Dale, Jul 25 2019 *)

a(n)=6^n-6*n \\ Charles R Greathouse IV, Jul 06 2017

a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3) for n > 2.
G.f.: (1-8*x+37*x^2)/((1-6*x)*(1-x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(5*x) - 6*x). - Elmo R. Oliveira, Sep 10 2024

A198397 a(n) = 7^n - 7*n.

Original entry on oeis.org

1, 0, 35, 322, 2373, 16772, 117607, 823494, 5764745, 40353544, 282475179, 1977326666, 13841287117, 96889010316, 678223072751, 4747561509838, 33232930569489, 232630513987088, 1628413597910323, 11398895185373010, 79792266297611861, 558545864083283860, 3909821048582987895

Offset: 0

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[7^n-7*n: n in [0..25]];
```

CoefficientList[Series[(1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n)=7^n-7*n \\ Charles R Greathouse IV, Jul 06 2017

a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3) for n > 2.
G.f.: (1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(6*x) - 7*x). - Elmo R. Oliveira, Sep 10 2024

A198398 a(n) = 8^n - 8n.

Original entry on oeis.org

1, 0, 48, 488, 4064, 32728, 262096, 2097096, 16777152, 134217656, 1073741744, 8589934504, 68719476640, 549755813784, 4398046510992, 35184372088712, 281474976710528, 2251799813685112, 18014398509481840, 144115188075855720, 1152921504606846816, 9223372036854775640

Offset: 0

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[8^n-8*n: n in [0..25]];
```

CoefficientList[Series[(1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{10,-17,8},{1,0,48},30] (* Harvey P. Dale, Mar 13 2023 *)

a(n)=8^n-8*n \\ Charles R Greathouse IV, Jul 06 2017

a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n > 2.
G.f.: (1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(7*x) - 8*x). - Elmo R. Oliveira, Sep 10 2024

A198399 a(n) = 9^n - 9*n.

Original entry on oeis.org

1, 0, 63, 702, 6525, 59004, 531387, 4782906, 43046649, 387420408, 3486784311, 31381059510, 282429536373, 2541865828212, 22876792454835, 205891132094514, 1853020188851697, 16677181699666416, 150094635296998959, 1350851717672991918, 12157665459056928621, 109418989131512359020

Offset: 0

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[9^n-9*n: n in [0..25]];
```

CoefficientList[Series[(1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n)=9^n-9*n \\ Charles R Greathouse IV, Jul 06 2017

a(0)=1, a(1)=0, a(2)=63, a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
G.f.: (1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(exp(8*x) - 9*x). - Elmo R. Oliveira, Sep 09 2024

A198400 a(n) = 10^n - 10*n.

Original entry on oeis.org

1, 0, 80, 970, 9960, 99950, 999940, 9999930, 99999920, 999999910, 9999999900, 99999999890, 999999999880, 9999999999870, 99999999999860, 999999999999850, 9999999999999840, 99999999999999830, 999999999999999820, 9999999999999999810, 99999999999999999800, 999999999999999999790

Offset: 0

Vincenzo Librandi, Oct 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A005803, A107583, A107584, A107585.

Magma
```
[10^n-10*n: n in [0..25]];
```

CoefficientList[Series[(1-12*x+101*x^2)/((1-10*x)*(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)

a(n)=10^n-10*n \\ Charles R Greathouse IV, Jul 06 2017

a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 2.
G.f.: (1-12*x+101*x^2)/((1-10*x)*(1-x)^2). - Vincenzo Librandi, Jul 06 2012
E.g.f.: exp(x)*(exp(9*x) - 10*x). - Elmo R. Oliveira, Aug 29 2024

A221907 a(n) = 5^n + 5*n.

Original entry on oeis.org

1, 10, 35, 140, 645, 3150, 15655, 78160, 390665, 1953170, 9765675, 48828180, 244140685, 1220703190, 6103515695, 30517578200, 152587890705, 762939453210, 3814697265715, 19073486328220, 95367431640725, 476837158203230, 2384185791015735, 11920928955078240, 59604644775390745

Offset: 0

Vincenzo Librandi, Mar 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A107585, A176916, A362555, A370557.

Magma
```
[5^n + 5*n: n in [0..30]];
```

I:=[1, 10, 35]; [n le 3 select I[n] else 7*Self(n-1)-11*Self(n-2)+5*Self(n-3): n in [1..30]];

Table[(5^n + 5 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + 3 x - 24 x^2)/((1 - x)^2 (1 - 5 x)), {x, 0, 30}], x]
LinearRecurrence[{7,-11,5},{1,10,35},30] (* Harvey P. Dale, Jul 23 2013 *)

a(n)=5^n+5*n \\ Charles R Greathouse IV, Apr 18 2013

G.f.: (1+3*x-24*x^2)/((1-x)^2*(1-5*x)).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
a(n) = A176916(n) - 1.
a(n) = 5*A362555(n) for n > 0. - Hugo Pfoertner, Mar 01 2024
E.g.f.: exp(x)*(exp(4*x) + 5*x). - Elmo R. Oliveira, Sep 10 2024

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

[(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016

Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]

Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024

cp's OEIS Frontend

A107583 a(n) = 3^n - 3*n.

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A107584 a(n) = 4^n - 4*n.

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A198396 a(n) = 6^n - 6*n.

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Magma

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A198397 a(n) = 7^n - 7*n.

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Magma

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A198398 a(n) = 8^n - 8n.

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Magma

Mathematica

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A198399 a(n) = 9^n - 9*n.

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Magma

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A198400 a(n) = 10^n - 10*n.

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A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.