A014291 Imaginary Rabbits: imaginary part of a(0)=i; a(1)=-i; a(n) = a(n-1) + i*a(n-2), with i = sqrt(-1).

1, -1, -1, -1, -2, -2, -1, 1, 5, 11, 18, 24, 25, 15, -13, -65, -142, -234, -313, -327, -199, 163, 838, 1840, 3041, 4079, 4279, 2639, -2042, -10802, -23841, -39519, -53155, -55989, -34982, 25544, 139225, 308895, 513547, 692655

Offset

0,5

Comments

Second differences give -a(n-2).

References

Charles Gely, "Lapins imaginaires et valeurs propres". Quadrature Quarterly #30.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-1).

Formula

a(n) = 2*a(n-1) - a(n-2) - a(n-4).

G.f.: (1-x)*(1-2*x)/(1-2*x+x^2+x^4).

a(n) = Im( -i*J(n, -i) + J(n-1, -i) ), where J(n,x) is the Jacobsthal polynomial, whose coefficient array is A128099, and i = sqrt(-1). - G. C. Greubel, Jun 15 2019

Mathematica

CoefficientList[Series[(2*x-1)*(x-1)/(x^4+x^2-2*x+1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 23 2012 *)

Prog

(Magma) I:=[1, -1, -1, -1]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) - Self(n-4): n in [1..50]]; // Vincenzo Librandi, Oct 23 2012
(PARI) my(x='x+O('x^50)); Vec((1-x)*(1-2*x)/(1-2*x+x^2+x^4)) \\ G. C. Greubel, Jun 13 2019
(Sage) ((1-x)*(1-2*x)/(1-2*x+x^2+x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019
(GAP) a:=[1, -1, -1, -1];; for n in [5..50] do a[n]:=2*a[n-1]-a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jun 13 2019

Crossrefs

Cf. A128099.

Sequence in context: A092450 A279629 A309575 * A136587 A136247 A370207

Adjacent sequences: A014288 A014289 A014290 * A014292 A014293 A014294

Keyword

sign,easy

Author

Olivier Gérard

Status

approved