This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A028860 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = -1, a(1) = 1. 14
 -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 (list; graph; refs; listen; history; text; internal format)
 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, RafaĆ 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 18 08:16 EST 2018. Contains 318219 sequences. (Running on oeis4.)