A168642 a(n) = (8*2^n + (-1)^n)/3 for n > 0; a(0) = 1.

1, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531

Offset: 0

Author

Klaus Brockhaus, Dec 01 2009

Keywords

• nonn
• easy

Comments

First differences of A085278.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A001045 (Jacobsthal sequence), A085278 (expansion of (1+2*x)^2/((1-2*x)*(1-x^2))).

Programs

Magma

[1] cat [ (8*2^n+(-1)^n)/3: n in [1..30] ];

Mathematica

Table[(8*2^n +(-1)^n)/3 - 2*Boole[n==0], {n, 0, 40}] (* or *) LinearRecurrence[{1,2}, {1,5,11}, 40] (* G. C. Greubel, Jul 28 2016; Feb 05 2021 *)

PARI

a(n)=([0,1; 2,1]^n*[1;5])[1,1] \\ Charles R Greathouse IV, Jul 29 2016

Sage

[1]+[(2^(n+3) +(-1)^n)/3 for n in (1..40)] # G. C. Greubel, Feb 05 2021

Formula

a(n) = A001045(n+3) for n > 0.

a(n) = a(n-1) + 2*a(n-2) for n > 2; a(0) = 1, a(1) = 5, a(2) = 11.

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

E.g.f.: (8*exp(2*x) - 6 + exp(-x))/3. - G. C. Greubel, Jul 28 2016