A097163 -id:A097163 - 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

A135520 a(n) = 4*a(n-2).

Original entry on oeis.org

2, 1, 8, 4, 32, 16, 128, 64, 512, 256, 2048, 1024, 8192, 4096, 32768, 16384, 131072, 65536, 524288, 262144, 2097152, 1048576, 8388608, 4194304, 33554432, 16777216, 134217728, 67108864, 536870912, 268435456, 2147483648, 1073741824

Offset: 0

Paul Curtz, Feb 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 4).

Cf. A097163, A097164.

[(5/4)*2^n+(3/4)*(-2)^n: n in [0..40]]; // Vincenzo Librandi, Jun 02 2011

A135520:=n->2^(n+(-1)^n); seq(A135520(n), n=0..50); # Wesley Ivan Hurt, Dec 13 2013

LinearRecurrence[{1,4,-4},{2,1,8},40] (* Harvey P. Dale, May 25 2012 *)
LinearRecurrence[{0, 4},{2, 1},32] (* Ray Chandler, Aug 03 2015 *)

a(n)=1<<(n+(-1)^n) \\ Charles R Greathouse IV, Jun 01 2011

From R. J. Mathar, corrected Apr 14 2008: (Start)
O.g.f.: (5/(1-2*x) + 3/(1+2*x))/4.
a(n) = (5*2^n + 3*(-2)^n)/4.
a(2*n)=2*A000302(n). a(2*n+1)=A000302(n). (End)
a(n) = A000079(n) terms swapped by pairs. - Paul Curtz, Apr 26 2011
a(n) = 2^(n+(-1)^n). - Wesley Ivan Hurt, Dec 13 2013
E.g.f.: (1/4)*(5*exp(2*x) + 3*exp(-2*x)). - G. C. Greubel, Oct 17 2016

More terms from R. J. Mathar, Feb 23 2008

A206374 a(n) = (7*4^n - 1)/3.

Original entry on oeis.org

2, 9, 37, 149, 597, 2389, 9557, 38229, 152917, 611669, 2446677, 9786709, 39146837, 156587349, 626349397, 2505397589, 10021590357, 40086361429, 160345445717, 641381782869, 2565527131477, 10262108525909, 41048434103637, 164193736414549, 656774945658197

Offset: 0

Brad Clardy, Feb 07 2012

First bisection of A062092 and A081253, second bisection of A097163. - Bruno Berselli, Feb 12 2012

Except a(0)=2, this is the 3rd row of table A178415. - Michel Marcus, Apr 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002450, A006666, A072197; A002042 (first differences), A178415, A347834.

Magma
```
[(7*4^n-1)/3 : n in [0..30]];
```

Table[(7(4^n) - 1)/3, {n, 0, 24}] (* Alonso del Arte, Feb 11 2012 *)
CoefficientList[Series[(2-x)/(1-5*x+4*x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-4},{2,9},30] (* Vincenzo Librandi, Mar 20 2012 *)

vector(20,n,(7*4^(n-1)-1)/3) \\ Derek Orr, Apr 12 2015

G.f.: (2-x)/(1-5*x+4*x^2). - Bruno Berselli, Feb 12 2012
a(n) = A083597(n)+1. - Bruno Berselli, Feb 12 2012
a(n) = 4*a(n-1)+1 for n>0, a(0)=2. - Bruno Berselli, Oct 22 2015
a(n) = 7*A002450(n) + 2. - Yosu Yurramendi, Jan 24 2017
A006666(a(n)) = 2*n+11 for n > 0. - Juan Miguel Barga Pérez, Jun 18 2020
a(n) = 5*a(n-1) - 4*a(n-2) for n >= 2. - Wesley Ivan Hurt, Jun 30 2020
a(n) = A178415(3, n) = A347834(4, n-1), arrays, for n >= 1.- Wolfdieter Lang, Nov 29 2021

A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

Original entry on oeis.org

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of A084221. a(n) = A097163(n+1)/4. Third binomial transform is A097165.
a(n+1) = 4*A097163(n). - Zerinvary Lajos, Mar 17 2008
See A133628 for an essentially identical sequence. - R. J. Mathar, Jun 08 2008

Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008

CoefficientList[Series[(1+3x)/((1-x)(1-4x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-4},{1,4,8},50] (* Harvey P. Dale, Jul 11 2023 *)

a(n) = 5*2^n/2 - (-2)^n/6 - 4/3;
a(n) = a(n-1) + 4a(n-2) - 4a(n-3).
G.f. ( 1+3*x ) / ( (x-1)*(2*x+1)*(2*x-1) ). - R. J. Mathar, Jul 06 2011

cp's OEIS Frontend

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

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A135520 a(n) = 4*a(n-2).

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A206374 a(n) = (7*4^n - 1)/3.

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A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

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