A132367 -id:A132367 - cp's OEIS frontend

A132357 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,4,14,41.

1, 4, 14, 41, 122, 364, 1093, 3280, 9842, 29525, 88574, 265720, 797161, 2391484, 7174454, 21523361, 64570082, 193710244, 581130733, 1743392200, 5230176602, 15690529805, 47071589414, 141214768240, 423644304721

Offset: 0

Paul Curtz, Nov 24 2007

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

First differences of A132353.

Cf. A129339.

LinearRecurrence[{3,0,-1,3},{1,4,14,41},50] (* Paolo Xausa, Dec 05 2023 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 3,-1,0,3]^n*[1;4;14;41])[1,1] \\ Charles R Greathouse IV, Oct 08 2016

O.g.f.: -(1+x+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/2)/(3*x-1)+(1/3)*(x-2)/(x^2-x+1)+(1/ 6)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) = (1/2)*3^(n+1) + (1/6)*(-1)^n - (2/3)*cos(Pi*n/3). Or, a(n) = (1/2)*3^(n+1) + (1/2)*[ -1; -1; 1; 1; 1; -1]. - Richard Choulet, Jan 02 2008
a(n+1) - 3a(n) = A132367(n+1). - Paul Curtz, Dec 02 2007
6*a(n) = (-1)^n +3^(n+2) -2*A057079(n+1). - R. J. Mathar, Oct 03 2021

A177702 Period 3: repeat [1, 1, 2].

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (2+sqrt(10))/3.
Decimal expansion of 112/999.
a(n) = A131534(n+2) = |A132419(n)| = |A132367(n)| = |A131556(n+2)|= |A122876(n)|.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A022003, A131534, A177703.

Magma
```
&cat[ [1, 1, 2]: k in [1..35] ];
```

seq(op([1, 1, 2]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{},120,{1,1,2}] (* or *) LinearRecurrence[{0,0,1},{1,1,2},120] (* Harvey P. Dale, Dec 19 2014 *)

a(n)=max(n%3,1) \\ Charles R Greathouse IV, Jul 17 2016

a(n) = a(n-3) for n > 2, with a(0) = 1, a(1) = 1, a(2) = 2.
G.f.: (1+x+2*x^2)/(1-x^3).
a(n) = 4/3 - cos(2*Pi*n/3)/3 - sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) = 1 + A022003(n). - Wesley Ivan Hurt, Jul 01 2016

A135266 Partial sums of A132357.

Original entry on oeis.org

0, 1, 5, 19, 60, 182, 546, 1639, 4919, 14761, 44286, 132860, 398580, 1195741, 3587225, 10761679, 32285040, 96855122, 290565366, 871696099, 2615088299, 7845264901, 23535794706, 70607384120, 211822152360, 635466457081

Offset: 0

Paul Curtz, Dec 02 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-1,4,-3).

Join[{0}, Table[(1/4)*3^(n + 1) - (1/12)*(-1)^n + (1/3)*Cos[Pi*n/3] - (Sqrt[3]/3)*Sin[Pi*n/3] - 1, {n, 1, 25}]] (* G. C. Greubel, Oct 07 2016 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -3,4,-1,-3,4]^n*[0;1;5;19;60])[1,1] \\ Charles R Greathouse IV, Oct 08 2016

a(n+1) - 3*a(n) = 0, 1, 2, 4, 3, 2,... (periodically extended with period length 6) = partial sums of A132367.
a(n) = (1/4)*3^(n+1) - (1/12)*(-1)^n + (1/3)*cos(Pi*n/3) - (sqrt(3)/3)*sin (Pi*n/3) - 1. Or, a(n) = (1/4)*3^(n+1) + (1/4)*[ -3; -5; -7; -5; -3; -1] for n>=0. - Richard Choulet, Jan 02 2008
O.g.f.: x*(1 +x +2*x^2)/((3*x-1)*(x+1)(x^2-x+1)*(x-1)). - R. J. Mathar, Jul 28 2008

Edited and extended by R. J. Mathar, Jul 28 2008

A132401 Period 8: repeat 0, 0, 1, 1, 2, -1, -1, -2.

Original entry on oeis.org

0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2, -1, -1, -2, 0, 0, 1, 1, 2

Offset: 0

Paul Curtz, Nov 12 2007

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

Cf. A132367.

a(n)=[0, 0, 1, 1, 2, -1, -1, -2][n%8+1] \\ Charles R Greathouse IV, Jul 13 2016

cp's OEIS Frontend

A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,4,14,41.

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A177702 Period 3: repeat [1, 1, 2].

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A135266 Partial sums of A132357.

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A132401 Period 8: repeat 0, 0, 1, 1, 2, -1, -1, -2.

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A132357 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,4,14,41.