A028860 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = -1, a(1) = 1.

-1, 1, 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432, 2627929735168

Offset

0,4

Comments

a(n+1) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 04 2014

(A002605, a(.+1)) is the canonical basis of the space of linear recurrent sequences with signature (2, 2), i.e., any sequence s(n) = 2(s(n-1) + s(n-2)) is given by s = s(0)*A002605 + s(1)*a(.+1). - M. F. Hasler, Aug 06 2018

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 924

Tanya Khovanova, Recursive Sequences

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

Formula

a(n) = 4*A028859(n-4), for n > 3.

From R. J. Mathar, Nov 27 2008: (Start)

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

a(n) = 3*A002605(n-1) - A002605(n). (End)

a(n) = det A, where A is the Hessenberg matrix of order n+1 defined by: A[i,j] = p(j - i + 1) (i <= j), A[i,j] = -1 (i = j + 1), A[i,j] = 0 otherwise, with p(i) = fibonacci(2i - 4). - Milan Janjic, May 08 2010, edited by M. F. Hasler, Aug 06 2018

a(n) = (2*sqrt(3) - 3)/6*(1 + sqrt(3))^n - (2*sqrt(3) + 3)/6*(1 - sqrt(3))^n. - Sergei N. Gladkovskii, Jul 18 2012

a(n) = 2*A002605(n-2) for n >= 2. - M. F. Hasler, Aug 06 2018

E.g.f.: exp(x)*(2*sqrt(3)*sinh(sqrt(3)*x) - 3*cosh(sqrt(3)*x))/3. - Franck Maminirina Ramaharo, Nov 11 2018

Maple

seq(coeff(series((3*x-1)/(1-2*x-2*x^2), x, n+1), x, n), n=0..30); # Muniru A Asiru, Aug 07 2018

Mathematica

(With a different offset) M = {{0, 2}, {1, 2}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}] (* Roger L. Bagula, May 29 2005 *)
LinearRecurrence[{2, 2}, {-1, 1}, 40] (* Harvey P. Dale, Dec 13 2012 *)
CoefficientList[Series[(-3 x + 1)/(2 x^2 + 2 x - 1), {x, 0, 27}], x] (* Robert G. Wilson v, Aug 07 2018 *)

Prog

(Haskell)
a028860 n = a028860_list !! n
a028860_list =
   -1 : 1 : map (* 2) (zipWith (+) a028860_list (tail a028860_list))
-- Reinhard Zumkeller, Oct 15 2011
(PARI) apply( A028860(n)=([2, 2; 1, 0]^n)[2, ]*[1, -1]~, [0..30]) \\ 15% faster than (A^n*[1, -1]~)[2]. - M. F. Hasler, Aug 06 2018
(GAP) a:=[-1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+2*a[n-2]; od; a; # Muniru A Asiru, Aug 07 2018
(Magma) I:=[-1, 1]; [n le 2 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2018
(SageMath)
A028860 = BinaryRecurrenceSequence(2, 2, -1, 1)
[A028860(n) for n in range(51)] # G. C. Greubel, Dec 08 2022

Crossrefs

Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898, A125145.

Sequence in context: A302919 A181329 A293007 * A152035 A026151 A025178

Adjacent sequences: A028857 A028858 A028859 * A028861 A028862 A028863

Keyword

sign,easy

Author

N. J. A. Sloane

Extensions

Edited by N. J. A. Sloane, Apr 11 2009

Edited and initial values added in definition by M. F. Hasler, Aug 06 2018

Status

approved