A084640 Generalized Jacobsthal numbers.

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105, 223696211, 447392425, 894784851, 1789569705, 3579139411

Offset: 0

Author

Paul Barry, Jun 06 2003

Keywords

• easy
• nonn

Comments

This is the sequence A(0,1;1,2;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A000975, A000225, A084639.

Programs

Haskell

a084640 n = a084640_list !! n
a084640_list = 0 : 1 : (map (+ 4) $
   zipWith (+) (map (* 2) a084640_list) (tail a084640_list))
-- Reinhard Zumkeller, May 23 2013

Magma

[5*2^n/3+(-1)^n/3-2: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

Mathematica

LinearRecurrence[{2,1,-2},{0,1,5},40] (* Harvey P. Dale, Oct 27 2015 *)

PARI

x='x+O('x^50); Vec(x*(1+3*x)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, Sep 26 2017

Formula

G.f.: x*(1+3*x)/((1-x^2)*(1-2*x)).

a(n) = a(n-1) + 2a(n-2) + 4, a(0)=0, a(1)=1.

a(n) = (5*2^n + (-1)^n - 6)/3.

a(n) = A001045(n+2) + 4*A000975(n-3).

a(n+1) - 2*a(n) = period 2: repeat 1, 3. - Paul Curtz, Apr 03 2008

Contribution from Paul Curtz, Dec 10 2009: (Start)

a(n+2) - a(n) = A020714(n).

Le the array D(n,k) of the first differences be defined via D(0,k) = a(k); D(n+1,k) = D(n,k+1)-D(n,k).

Then D(n,n) = 4*A131577(n); D(1,k) = A084214(k+1); D(2,k) = A115102(k-1) for k>0; D(3,k) = (-1)^(k+1)*A083581(k). (End)

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010