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, 2541865828330, 7625597484986, 22876792454962

Offset

0,1

Comments

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

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2095

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

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).

Formula

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)

Mathematica

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

Prog

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

Crossrefs

Apart from leading term, same as A084182.

Cf. A000244, A046717, A105723.

Sequence in context: A341680 A213338 A309753 * A151364 A200949 A001885

Adjacent sequences: A102342 A102343 A102344 * A102346 A102347 A102348

Keyword

easy,nonn

Author

Graeme McRae, Feb 16 2005

Status

approved