A084182 -id:A084182 - cp's OEIS frontend

A102345 a(n) = 3^n + (-1)^n.

2, 2, 10, 26, 82, 242, 730, 2186, 6562, 19682, 59050, 177146, 531442, 1594322, 4782970, 14348906, 43046722, 129140162, 387420490, 1162261466, 3486784402, 10460353202, 31381059610, 94143178826, 282429536482, 847288609442

Offset: 0

Graeme McRae, Feb 16 2005

a(n) = A105723(n) + 2*(-1)^n; (a(n) + A105723(n))/2 = A000244(n). - Reinhard Zumkeller, Apr 18 2005

Michael De Vlieger, Table of n, a(n) for n = 0..2095
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Apart from leading term, same as A084182.

Cf. A000244, A046717, A105723.

Table[3^n+(-1)^n,{n,0,30}] (* or *) LinearRecurrence[{2,3},{2,2},30] (* Harvey P. Dale, Jun 19 2016 *)

[lucas_number2(n,2,-3) for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(n) = 2*a(n-1) + 3*a(n-2).
From Elmo R. Oliveira, Dec 18 2023: (Start)
G.f.: 2*(1-x)/((1+x)*(1-3*x)).
E.g.f.: exp(-x) + exp(3*x).
a(n) = 2*A046717(n). (End)

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A084181 2^n+(-2)^n-(-1)^n.

Original entry on oeis.org

1, 1, 7, 1, 31, 1, 127, 1, 511, 1, 2047, 1, 8191, 1, 32767, 1, 131071, 1, 524287, 1, 2097151, 1, 8388607, 1, 33554431, 1, 134217727, 1, 536870911, 1, 2147483647, 1, 8589934591, 1, 34359738367, 1, 137438953471, 1, 549755813887, 1, 2199023255551

Offset: 0

Paul Barry, May 19 2003

Binomial transform is A084182.

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

Cf. A083420.

LinearRecurrence[{-1,4,4},{1,1,7},50] (* or *) Riffle[ LinearRecurrence[ {5,-4},{1,7},30],1] (* Harvey P. Dale, Jan 02 2019 *)

a(n)=2^n+(-2)^n-(-1)^n;
G.f.: (1+2x+4x^2)/((1+x)(1+2x)(1-2x));
E.g.f.: exp(2x)-exp(-x)+exp(-2x).

cp's OEIS Frontend

A102345 a(n) = 3^n + (-1)^n.

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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A084181 2^n+(-2)^n-(-1)^n.

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