A080040 a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.

2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480, 23168, 63296, 172928, 472448, 1290752, 3526400, 9634304, 26321408, 71911424, 196465664, 536754176, 1466439680, 4006387712, 10945654784, 29904084992, 81699479552, 223207129088, 609813217280, 1666040692736, 4551707820032

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003

Keywords

• easy
• nonn

Comments

The Lucas sequence V_n(2,-2). - Jud McCranie, Mar 02 2012

The signed version 2, -2, 8, -20, 56, -152, 416, -1136, 3104, -8480, 23168, ... is the Lucas sequence V(-2,-2). - R. J. Mathar, Jan 08 2013

After a(2) equals round((1+sqrt(3))^n) = 1, 3, 7, 20, 56, 152, ... - Jeremy Gardiner, Aug 11 2013

Also the number of independent vertex sets and vertex covers in the n-sunlet graph. - Eric W. Weisstein, Sep 27 2017

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO], 2015-2017.
• 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.
• D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
• Tanya Khovanova, Recursive Sequences
• D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See V_n.
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Eric Weisstein's World of Mathematics, Sunlet Graph
• Eric Weisstein's World of Mathematics, Vertex Cover
• Index entries for linear recurrences with constant coefficients, signature (2,2).
• Index entries for Lucas sequences (2,-2).

Crossrefs

Cf. A002531, A002605, A028859, A030195, A083337, A106435, A108898, A125145.

Equals 2*A026150.

Programs

Haskell

a080040 n = a080040_list !! n
a080040_list =
   2 : 2 : map (* 2) (zipWith (+) a080040_list (tail a080040_list))
-- Reinhard Zumkeller, Oct 15 2011

Magma

a:=[2,2]; [n le 2 select a[n] else 2*Self(n-1) + 2*Self(n-2):n in [1..27]]; Marius A. Burtea, Jan 20 2020

Magma

R:=PowerSeriesRing(Rationals(), 27); Coefficients(R!( (2-2*x)/(1-2*x-2*x^2))); // Marius A. Burtea, Jan 20 2020

Mathematica

CoefficientList[Series[(2 - 2 t)/(1 - 2 t - 2 t^2), {t, 0, 30}], t]
With[{c = {2, 2}}, LinearRecurrence[c, c, 20]] (* Harvey P. Dale, Apr 24 2016 *)
Round @ Table[LucasL[n, Sqrt[2]] 2^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
Table[(1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 27 2017 *)

PARI

a(n)=([0,1; 2,2]^n*[2;2])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

Sage

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(2,2,2,2, lambda n: 0); [next(it) for i in range(27)] # Zerinvary Lajos, Jul 16 2008

Sage

[lucas_number2(n,2,-2) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009

Formula

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

a(n) = (1+sqrt(3))^n + (1-sqrt(3))^n.

a(n) = 2*A026150(n). - Philippe Deléham, Nov 19 2008

G.f.: G(0), where G(k) = 1 + 1/(1 - x*(3*k-1)/(x*(3*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013

a(n) = 2*2^floor(n/2)*A002531(n). - Ralf Stephan, Sep 08 2013

a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 3*x^2) )^n for n >= 1. - Peter Bala, Jun 29 2015

E.g.f.: 2*exp(x)*cosh(sqrt(3)*x). - Stefano Spezia, Mar 02 2024