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

Author

N. J. A. Sloane

Keywords

• sign
• easy

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).

Crossrefs

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

Programs

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

Haskell

a028860 n = a028860_list !! n
a028860_list =
   -1 : 1 : map (* 2) (zipWith (+) a028860_list (tail a028860_list))
-- Reinhard Zumkeller, Oct 15 2011

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

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 *)

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

SageMath

A028860 = BinaryRecurrenceSequence(2,2,-1,1)
[A028860(n) for n in range(51)] # G. C. Greubel, Dec 08 2022

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

Extensions

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

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