A136326 -id:A136326 - cp's OEIS frontend

A140966 a(n) = (5 + (-2)^n)/3.

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

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

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A330246 a(n) = 4^(n+1) + (4^n-1)/3.

Original entry on oeis.org

4, 17, 69, 277, 1109, 4437, 17749, 70997, 283989, 1135957, 4543829, 18175317, 72701269, 290805077, 1163220309, 4652881237, 18611524949, 74446099797, 297784399189, 1191137596757, 4764550387029, 19058201548117, 76232806192469, 304931224769877, 1219724899079509

Offset: 0

Vincenzo Librandi, Jan 09 2020

After 4, these numbers are the third column of the rectangular array in A238475.

Similar to A272743.
Cf. A000302, A002450, A052909, A178415, A237930, A238475, A347834.
Together with 1: first bisection of A136326.

Magma
```
[4^(n+1)+(4^n-1)/3: n in [0..30]];
```

Table[(4^(n + 1) + (4^n - 1) / 3), {n, 0, 30}]

G.f.: (4 - 3*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n) = 4*a(n-1) + 1 for n > 0.
a(n) = (13*4^n -1)/3, for n >= 0. - Wolfdieter Lang, Sep 16 2021
a(n) = A178415(5, n) = A347834(7, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

cp's OEIS Frontend

A140966 a(n) = (5 + (-2)^n)/3.

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