A090017 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.

0, 1, 4, 18, 80, 356, 1584, 7048, 31360, 139536, 620864, 2762528, 12291840, 54692416, 243353344, 1082798208, 4817899520, 21437194496, 95384577024, 424412697088, 1888419942400, 8402505163776, 37386860539904, 166352452487168

Offset

0,3

Comments

Starting with "1" = INVERT transform of A007482: (1, 3, 11, 39, 139, ...). - Gary W. Adamson, Aug 06 2010

This is the Lucas sequence U(4,-2). - Bruno Berselli, Jan 09 2013

Lower left term in matrix powers of [(1,5); (1,3)]. Convolved with (1, 2, 0, 0, 0, ...) the result is A164549: (1, 6, 26, 116, ...). - Gary W. Adamson, Aug 10 2016

For n>0, a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Wikipedia, Lucas sequence: Specific names.

Index entries for sequences related to Chebyshev polynomials.

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

Formula

G.f.: x/(1 - 4*x - 2*x^2).

a(n) = (-i*sqrt(2))^(n-1) U(n-1, i*sqrt(2)) where U is the Chebyshev polynomial of the second kind and i^2 = -1.

a(n) = ((2+sqrt(6))^n - (2-sqrt(6))^n)/(2 sqrt(6)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009, Jan 07 2009

a(n+1) = Sum_{k=0..n} A099089(n,k)*2^k. - Philippe Deléham, Nov 21 2011

From Ilya Gutkovskiy, Aug 22 2016: (Start)

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

Number of zeros in substitution system {0 -> 11, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 1101111011111101111011 -> ...). (End)

Mathematica

a[n_Integer] := (-I Sqrt[2])^(n - 1) ChebyshevU[ n - 1, I Sqrt[2] ]
a[n_]:=(MatrixPower[{{1, 5}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
t={0, 1}; Do[AppendTo[t, 4*t[[-1]]+2*t[[-2]]], {n, 2, 23}]; t (* or *) LinearRecurrence[{4, 2}, {0, 1}, 24] (* Indranil Ghosh, Feb 21 2017 *)

Prog

(Sage) [lucas_number1(n, 4, -2) for n in range(0, 23)] # Zerinvary Lajos, Apr 23 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2011
(PARI) Vec(x/(1-4*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Oct 12 2011

Crossrefs

Cf. A007070, A084059, A007482.

Cf. A164549.

Sequence in context: A177755 A037965 A045902 * A257390 A104631 A106391

Adjacent sequences: A090014 A090015 A090016 * A090018 A090019 A090020

Keyword

nonn,easy

Author

Paul Barry, Nov 19 2003

Extensions

Edited by Stuart Clary, Oct 25 2009

Status

approved