A084170 a(n) = (5*2^n + (-1)^n - 3)/3.

1, 2, 6, 12, 26, 52, 106, 212, 426, 852, 1706, 3412, 6826, 13652, 27306, 54612, 109226, 218452, 436906, 873812, 1747626, 3495252, 6990506, 13981012, 27962026, 55924052, 111848106, 223696212, 447392426, 894784852, 1789569706, 3579139412

Offset: 0

Author

Paul Barry, May 18 2003

Keywords

• easy
• nonn

Comments

Original name of this sequence: Generalized Jacobsthal numbers.

Links

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

Crossrefs

Cf. A000975, A002450, A026644, A146882, A169969.

Cf. A000225 (Mersenne numbers), A001045 (Jacobsthal numbers).

Programs

Magma

[(5*2^n +(-1)^n)/3 -1: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

Mathematica

LinearRecurrence[{2,1,-2},{1,2,6},40] (* or *) Table[(5*2^n+(-1)^n-3)/3,{n,0,40}] (* Harvey P. Dale, Jan 29 2012 *)

PARI

a(n)=(5*2^n)\/3-1 \\ Charles R Greathouse IV, Jul 01 2011

SageMath

[(2/3)*(5*2^(n-1) -1 -(n%2)) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.

a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=1, a(1)=2.

G.f.: (1+x^2)/((1+x)*(1-x)*(1-2*x)).

E.g.f.: 5*exp(2*x)/3 - exp(x) + exp(-x)/3.

a(n+1) = A000975(n+2) + A000975(n).

a(2*n+1) - 2 = 10*A000975(n).

a(2*n+2) - 6 = 20*A000975(n).

a(n+2*k) - a(n) = 5*A002450(k)*2^n = A146882(k-1)*2^n, k >= 0. - Paul Curtz, Jun 15 2011

From Yosu Yurramendi, Jul 05 2016: (Start)

a(n) = A169969(2n) - 1, n >= 1; a(n) = 3*2^(n-1) - 1 + A169969(2n-7), n >= 5.

a(n+3) = 15*2^n - 2 - a(n), n >= 0, a(0)=1, a(1)=2, a(2)=6.

a(n) + A026644(n) = 3*2^n - 2, n >= 1.

a(n+3) = 3*2^(n+2) + A026644(n), n >= 1. (End)

a(n) = A000225(n+1) - A001045(n). - Yuchun Ji, Mar 17 2020