A135351 -id:A135351 - cp's OEIS frontend

A155980 First differences of A135351.

0, 2, -2, 3, -1, 5, 3, 13, 19, 45, 83, 173, 339, 685, 1363, 2733, 5459, 10925, 21843, 43693, 87379, 174765, 349523, 699053, 1398099, 2796205, 5592403, 11184813, 22369619, 44739245, 89478483, 178956973, 357913939, 715827885, 1431655763, 2863311533, 5726623059

Offset: 0

Paul Curtz, Feb 01 2009

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A135351.

LinearRecurrence[{1,2},{0,2,-2,3},50] (* Paolo Xausa, Nov 07 2023 *)

G.f.: (x^2-4*x+2)*x/((1-2*x)*(x+1)). - Alois P. Heinz, Jan 21 2021

More terms from Alois P. Heinz, Jan 21 2021

A163834 a(n) = (4^n + 5)/3.

Original entry on oeis.org

2, 3, 7, 23, 87, 343, 1367, 5463, 21847, 87383, 349527, 1398103, 5592407, 22369623, 89478487, 357913943, 1431655767, 5726623063, 22906492247, 91625968983, 366503875927, 1466015503703, 5864062014807, 23456248059223, 93824992236887, 375299968947543

Offset: 0

Juri-Stepan Gerasimov, Aug 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000027, A135351, A140966, A153643.

Table[(4^n + 5)/3, {n, 0, 50}] (* G. C. Greubel, Aug 05 2017 *)
LinearRecurrence[{5,-4},{2,3},30] (* Harvey P. Dale, Jun 14 2023 *)

x='x+O('x^50); concat([0], Vec((2-7*x)/((4*x-1)*(x-1)))) \\ G. C. Greubel, Aug 05 2017

a(n) = (4^n + 5)/3 = A135351(2*n+1) = A140966(2*n) = A153643(2*n).
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (2-7*x)/((4*x-1)*(x-1)).
a(n+1) - a(n) = A000302(n).
E.g.f.: (1/3)*(5*exp(x) + exp(4*x)). - G. C. Greubel, Aug 05 2017

Offset set to 0 by R. J. Mathar, Aug 06 2009

A099754 a(n) = (3^n +1)/2 + 2^n.

Original entry on oeis.org

2, 4, 9, 22, 57, 154, 429, 1222, 3537, 10354, 30549, 90622, 269817, 805354, 2407869, 7207222, 21588897, 64701154, 193972389, 581655022, 1744440777, 5232273754, 15694724109, 47079978022, 141231545457, 423677859154, 1271000023029

Offset: 0

Miklos Kristof, Nov 11 2004

Let b(0)=1, b(n) = A005578(n-1) = {1,1,2,3,6,11,22,43,86,171,342, ...} then a(n) = Sum_{k=0..n+1} C(n+1,k)*b(k).

Binomial transform of A135351. - R. J. Mathar, Aug 05 2009

			a(6) = (3^6+1)/2 + 2^6 = 365+64 = 429.
a(6) = 1 + 7*1 + 21*1 + 35*2 + 35*3 + 21*6 + 7*11 + 1*22 = 429.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yilmaz Simsek, New families of special numbers for computing negative order Euler numbers, arXiv:1604.05601 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A005578.

List([0..30], n-> (3^n +2^(n+1) +1)/2); # G. C. Greubel, Sep 03 2019

[(3^n +2^(n+1) +1)/2: n in [0..30]]; // G. C. Greubel, Sep 03 2019

seq((3^n +2^(n+1) +1)/2, n=0..30); # G. C. Greubel, Sep 03 2019

Table[(3^n +2^(n+1) +1)/2, {n,0,30}] (* G. C. Greubel, Sep 03 2019 *)
LinearRecurrence[{6,-11,6},{2,4,9},30] (* Harvey P. Dale, May 23 2021 *)

a(n) = (3^n+1)/2 + 2^n; \\ Michel Marcus, Aug 15 2013

[(3^n +2^(n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Sep 03 2019

a(n) = (3^n + 2^(n+1) + 1)/2.
G.f.: (2-8*x+7*x^2)/((1-x)*(1-2*x)*(1-3*x)). - Jaume Oliver Lafont, Mar 06 2009
a(n) = A007051(n) + A000079(n). - Michel Marcus, Aug 15 2013
E.g.f.: (exp(x) + 2*exp(2*x) + exp(3*x))/2. - G. C. Greubel, Sep 03 2019

Corrected and extended by T. D. Noe, Nov 07 2006

A171382 a(n) = (22^n+7(-1)^n)/3.

Original entry on oeis.org

3, -1, 5, 3, 13, 19, 45, 83, 173, 339, 685, 1363, 2733, 5459, 10925, 21843, 43693, 87379, 174765, 349523, 699053, 1398099, 2796205, 5592403, 11184813, 22369619, 44739245, 89478483, 178956973, 357913939, 715827885, 1431655763, 2863311533

Offset: 0

Klaus Brockhaus, Dec 07 2009

a(n) = A155980(n+2).
a(n) = A135351(n+3)-A135351(n+2).
Second binomial transform of a signed version of A005032 preceded by 3.
Inverse binomial transform of A008776 preceded by 3.

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A155980 (First differences of A135351), A135351 ((2^n+3-7*(-1)^n+3*0^n)/6), A005032 (7*3^n), A008776 (2*3^n).

Magma
```
[ (2*2^n+7*(-1)^n)/3: n in [0..32] ];
```

Nest[Append[#,Last[#]+2#[[-2]]]&,{3,-1},40]  (* Harvey P. Dale, Apr 07 2011 *)

a(n) = a(n-1)+2*a(n-2) for n > 1; a(0) = 3, a(1) = -1.
a(n) = 2^n-a(n-1) for n > 0; a(0) = 3.
G.f.: (3-4*x)/((1+x)*(1-2*x)).

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A155980 First differences of A135351.

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A163834 a(n) = (4^n + 5)/3.

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A099754 a(n) = (3^n +1)/2 + 2^n.

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A171382 a(n) = (2*2^n+7*(-1)^n)/3.

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A171382 a(n) = (22^n+7(-1)^n)/3.