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A090017 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1. 23
0, 1, 4, 18, 80, 356, 1584, 7048, 31360, 139536, 620864, 2762528, 12291840, 54692416, 243353344, 1082798208, 4817899520, 21437194496, 95384577024, 424412697088, 1888419942400, 8402505163776, 37386860539904, 166352452487168 (list; graph; refs; listen; history; text; internal format)
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

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 *)

PROG

(Sage) [lucas_number1(n, 4, -2) for n in xrange(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

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Last modified December 9 20:00 EST 2016. Contains 278986 sequences.