A153643 Jacobsthal numbers A001045 incremented by 2.

2, 3, 3, 5, 7, 13, 23, 45, 87, 173, 343, 685, 1367, 2733, 5463, 10925, 21847, 43693, 87383, 174765, 349527, 699053, 1398103, 2796205, 5592407, 11184813, 22369623, 44739245, 89478487, 178956973, 357913943, 715827885, 1431655767, 2863311533, 5726623063

Offset: 0

Author

Paul Curtz, Dec 30 2008

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A001045, A007395, A010684, A078008, A128209.

Programs

GAP

a:=[2,3,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Apr 02 2019

Magma

I:=[2,3,3]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..40]]; // G. C. Greubel, Apr 02 2019

Mathematica

LinearRecurrence[{1,2},{0,1}, 40] + 2 (* Harvey P. Dale, May 26 2014 *)
LinearRecurrence[{2,1,-2},{2,3,3}, 40] (* Georg Fischer, Apr 02 2019 *)

PARI

my(x='x+O('x^40)); Vec( (2-x-5*x^2)/((1-x^2)*(1-2*x)) ) \\ G. C. Greubel, Apr 02 2019

Python

def A153643(n): return ((1<Chai Wah Wu, Apr 18 2025

Sage

((2-x-5*x^2)/((1-x^2)*(1-2*x))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 02 2019

Formula

a(n) = 2 + A001045(n) = A001045(n) + A007395(n) = 1 + A128209(n).

a(n) - A010684(n) = A078008(n), first differences of A001045. - Paul Curtz, Jan 17 2009

G.f.: (2 - x - 5*x^2)/((1+x)*(1-x)*(1-2*x)). - R. J. Mathar, Jan 23 2009

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n >= 3. - Andrew Howroyd, Feb 26 2018

Extensions

Edited and extended by R. J. Mathar, Jan 23 2009