A084059 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=2.

1, 2, 10, 44, 196, 872, 3880, 17264, 76816, 341792, 1520800, 6766784, 30108736, 133968512, 596091520, 2652303104, 11801395456, 52510188032, 233643543040, 1039594548224, 4625665278976, 20581850212352, 91578731407360

Offset

0,2

Comments

2*A084059 is the Lucas sequence V(4,-2). - Bruno Berselli, Jan 09 2013

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,2).

Formula

E.g.f.: exp(2*x)*cosh(sqrt(6)*x).

a(n) = ((2+sqrt(6))^n + (2-sqrt(6))^n)/2. - Paul Barry, May 13 2003

a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*2^(n-k)*3^k. - Paul Barry, Jan 15 2007

G.f.: (1-2*x)/(1-4*x-2*x^2). - Philippe Deléham, Sep 07 2009

a(n) = A090017(n+1) - 2*A090017(n). - R. J. Mathar, Apr 05 2011

a(n) = Sum_{k=0..n} A201730(n,k)*5^k. - Philippe Deléham, Dec 06 2011

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-2)/(x*(3*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013

a(n) = (-i)^n*2^(n/2)*ChebyshevT(n, i*sqrt(2)) = 2^((n-2)/2)*Lucas(n, 2*sqrt(2)). - G. C. Greubel, Jan 03 2020

Maple

seq(simplify(2^(n/2)*(-I)^n*ChebyshevT(n, I*sqrt(2))), n = 0..30); # G. C. Greubel, Jan 03 2020

Mathematica

Table[(-I)^n*2^(n/2)*ChebyshevT[n, I*Sqrt[2]], {n, 0, 30}] (* G. C. Greubel, Jan 03 2020 *)

Prog

(Sage) [lucas_number2(n, 4, -2)/2 for n in range(0, 30)] # Zerinvary Lajos, May 14 2009
(Magma) [n le 2 select n else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 05 2011
(PARI) Vec((1-2*x)/(1-4*x-2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016
(PARI) vector(31, n, round((-I)^(n-1)*2^((n-1)/2)*polchebyshev(n-1, 1, I*sqrt(2))) ) \\ G. C. Greubel, Jan 03 2020
(GAP) a:=[1, 2];; for n in [3..30] do a[n]:=4*a[n-1]+2*a[n-2]; od; a; # G. C. Greubel, Jan 03 2020

Crossrefs

Cf. A090017, A084120 (binom. transf.), A002533 (inv. bin. transf).

Sequence in context: A068551 A099919 A100397 * A339642 A084609 A105485

Adjacent sequences: A084056 A084057 A084058 * A084060 A084061 A084062

Keyword

nonn,easy

Author

Paul Barry, May 10 2003

Status

approved