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

0,4

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).

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

Mathematica

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

Prog

(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
(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
(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
(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

Crossrefs

Cf. A083580.

Sequence in context: A163318 A129362 A301981 * A335714 A215112 A340948

Adjacent sequences: A083576 A083577 A083578 * A083580 A083581 A083582

Keyword

nonn,easy

Author

Paul Barry, May 01 2003

Status

approved