A143055 The real part of complex sequence: a(n) = a(n-1) + (1+i)*a(n-2).

0, 1, 1, 2, 3, 4, 5, 4, -1, -16, -51, -124, -265, -520, -955, -1652, -2689, -4080, -5635, -6668, -5433, 1896, 22965, 72028, 174095, 370496, 725101, 1328452, 2292823, 3722904, 5631525, 7743404, 9086879, 7208304, -3246371, -32945004, -101726745

Offset

1,4

Comments

The (absolute) ratio approaches 1.744900645213449...

Links

Robert Israel, Table of n, a(n) for n = 1..4120

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

Formula

a(n) = Re(b(n)) where b(1) = 0, b(2) = 1, b(n) = b(n-1)+(1+i)*b(n-2).

From R. J. Mathar, Oct 24 2008: (Start)

G.f.: x^2*(1 - x - x^2)/(1-2*x-x^2+2*x^3+2*x^4).

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 2*a(n-4). (End)

Example

The imaginary part is: {0, 0, 0, 1, 2, 5, 10, 19, 34, 57, 90, 131, 170, 177, 82, -261, -1134, -3047, -6870, -13997, -26502, -47167, -79102, -124373, -180510, -232855, -239270, -101629, 384202, 1611025, 4288050,..}.

Maple

f:= gfun:-rectoproc({-a(n+4)+2*a(n+3)+a(n+2)-2*a(n+1)-2*a(n), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Jul 17 2016

Mathematica

Clear[a, n]; a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + (1 + I)*a[n - 2]; Table[Re[a[n]], {n, 0, 30}]
CoefficientList[Series[x (1 - x - x^2) / (1 - 2 x - x^2 + 2 x^3 + 2 x^4), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 18 2016 *)
LinearRecurrence[{2, 1, -2, -2}, {0, 1, 1, 2}, 40] (* Harvey P. Dale, Oct 26 2019 *)

Prog

(Magma) I:=[0, 1, 1, 2]; [n le 4 select I[n] else 2*Self(n-1) +Self(n-2)-2*Self(n-3)-2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 18 2016

Crossrefs

Sequence in context: A125746 A337376 A237985 * A339388 A271751 A017890

Adjacent sequences: A143052 A143053 A143054 * A143056 A143057 A143058

Keyword

sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 13 2008

Extensions

Edited by Robert Israel, Jul 17 2016

Status

approved