A242329 -id:A242329 - cp's OEIS frontend

A253208 a(n) = 4^n + 3.

4, 7, 19, 67, 259, 1027, 4099, 16387, 65539, 262147, 1048579, 4194307, 16777219, 67108867, 268435459, 1073741827, 4294967299, 17179869187, 68719476739, 274877906947, 1099511627779, 4398046511107, 17592186044419, 70368744177667, 281474976710659

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226807.

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A141725, A226807.

Cf. Numbers of the form k^n+k-1: A000057 (k=2), A168607 (k=3), this sequence (k=4), A242329 (k=5), A253209 (k=6), A253210 (k=7), A253211 (k=8), A253212 (k=9), A253213 (k=10).

Magma
```
[4^n+3: n in [0..30]];
```

Table[4^n + 3, {n, 0, 30}] (* or *) CoefficientList[Series[(4 - 13 x) / ((1 - x) (1 - 4 x)), {x, 0, 40}], x]

a(n)=4^n+3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (4 - 13*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
From Elmo R. Oliveira, Nov 14 2023: (Start)
a(n) = 4*a(n-1) - 9 with a(0) = 4.
E.g.f.: exp(4*x) + 3*exp(x). (End)

A178676 a(n) = 5^n + 5.

Original entry on oeis.org

6, 10, 30, 130, 630, 3130, 15630, 78130, 390630, 1953130, 9765630, 48828130, 244140630, 1220703130, 6103515630, 30517578130, 152587890630, 762939453130, 3814697265630, 19073486328130, 95367431640630, 476837158203130, 2384185791015630, 11920928955078130

Offset: 0

Vincenzo Librandi, Dec 25 2010

			G.f. = 6 + 10*x + 30*x^2 + 130*x^3 + 630*x^4 + 3130*x^5 + 15630*x^6 + ... - _Michael Somos_, Jan 28 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A034474, A178671, A242328, A242329.

List([0..30], n -> 5^n +5); # G. C. Greubel, Jan 27 2019

Magma
```
[5^n+5: n in [0..35]];
```

Table[5^n + 5, {n, 0, 40}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n)=5^n+5 \\ Charles R Greathouse IV, Oct 07 2015

[5^n+5 for n in range(40)] # G. C. Greubel, Jan 27 2019

a(n) = 5*(a(n-1) - 4) with a(0) = 6.
G.f.: (6-26*x)/((1-5*x)*(1-x)). - R. J. Mathar, Jan 05 2011
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Vincenzo Librandi, Sep 30 2013
E.g.f.: exp(5*x) + 5*exp(x). - G. C. Greubel, Jan 27 2019
a(n) = A000351(n)+5 = A034474(n)+4 = A242328(n)+3 = A242329(n)+1. - Elmo R. Oliveira, Dec 06 2023

A242328 a(n) = 5^n + 2.

Original entry on oeis.org

3, 7, 27, 127, 627, 3127, 15627, 78127, 390627, 1953127, 9765627, 48828127, 244140627, 1220703127, 6103515627, 30517578127, 152587890627, 762939453127, 3814697265627, 19073486328127, 95367431640627, 476837158203127, 2384185791015627, 11920928955078127

Offset: 0

Vincenzo Librandi, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A034474, A132079, A178676, A242329.

Magma
```
[5^n+2: n in [0..30]];
```

Table[5^n + 2, {n, 0, 30}] (* or *) CoefficientList[Series[(3 - 11 x)/((1 - x) (1 - 5 x)), {x, 0, 30}], x]
LinearRecurrence[{6,-5},{3,7},30] (* Harvey P. Dale, Jun 30 2022 *)

G.f.: (3-11*x)/((1-x)*(1-5*x)).
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Elmo R. Oliveira, Dec 04 2023: (Start)
a(n) = A000351(n) + 2.
a(n) = 5*a(n-1) - 8 with a(0) = 3.
E.g.f.: exp(5*x) + 2*exp(x). (End)

A178671 a(n) = 5^n - 5.

Original entry on oeis.org

-4, 0, 20, 120, 620, 3120, 15620, 78120, 390620, 1953120, 9765620, 48828120, 244140620, 1220703120, 6103515620, 30517578120, 152587890620, 762939453120, 3814697265620, 19073486328120, 95367431640620, 476837158203120, 2384185791015620, 11920928955078120

Offset: 0

Vincenzo Librandi, Dec 25 2010

			a(n) = A178676(n)-10 = A242329(n)-9 = A242328(n)-7 = A034474(n)-6 = A000351(n)-5. - _Elmo R. Oliveira_, Dec 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A034474, A178676, A242328, A242329.

List([0..30], n -> 5^n-5); # G. C. Greubel, Jan 28 2019

Magma
```
[5^n-5: n in [0..25]];
```

5^Range[0,30]-5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{-4,0},30] (* Harvey P. Dale, Aug 23 2024 *)

vector(30, n, n--; 5^n-5) \\ G. C. Greubel, Jan 28 2019

[5^n-5 for n in range(30)] # G. C. Greubel, Jan 28 2019

a(n) = 5*a(n-1) + 20 with a(0) = -4.
From R. J. Mathar, Jan 03 2011: (Start)
G.f.: 4*(-1+6*x)/((1-5*x)*(1-x)).
a(n) = 4*A104891(n-1), n > 0. (End)
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Vincenzo Librandi, Jan 25 2013
E.g.f.: exp(5*x) - 5*exp(x). - G. C. Greubel, Jan 28 2019

cp's OEIS Frontend

A253208 a(n) = 4^n + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A178676 a(n) = 5^n + 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A242328 a(n) = 5^n + 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A178671 a(n) = 5^n - 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula