A083579 Generalized Jacobsthal numbers.

0, 1, 1, 4, 8, 19, 39, 82, 166, 337, 677, 1360, 2724, 5455, 10915, 21838, 43682, 87373, 174753, 349516, 699040, 1398091, 2796191, 5592394, 11184798, 22369609, 44739229, 89478472, 178956956, 357913927, 715827867, 1431655750, 2863311514

Offset: 0

Author

Paul Barry, May 01 2003

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 (3,-1,-3,2).

Crossrefs

Cf. A083580.

Programs

GAP

a:=[0,1,1,4];; for n in [5..40] do a[n]:=3*a[n-1]-a[n-2]-3*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, May 24 2019

Magma

I:=[0,1,1,4]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2) - 3*Self(n-3)+2*Self(n-4): n in [1..40]]; // G. C. Greubel, May 25 2019

Mathematica

LinearRecurrence[{3,-1,-3,2}, {0,1,1,4}, 40] (* G. C. Greubel, May 25 2019 *)

PARI

concat(0, Vec(x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4) + O(x^40))) \\ G. C. Greubel, May 25 2019

Sage

(x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

Formula

a(n) = (2^(n+3) - 5*(-1)^n - 3*(2*n+1))/12.

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

G.f.: x*(1 - 2*x + 2*x^2)/(1 - 3*x + x^2 + 3*x^3 - 2*x^4). - Colin Barker, Jan 16 2012