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A090965 a(n) = 8*a(n-1) - 4*a(n-2), where a(0) = 1, a(1) = 4. 10
1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, 2003193856, 14952042496, 111603564544, 833020346368, 6217748512768, 46409906716672, 346408259682304, 2585626450591744, 19299378566004736 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (8,-4).

FORMULA

a(n) = Sum_{k>=0} binomial(2*n, 2*k)*3^k = Sum_{k>=0} A086645(n, k)*3^k.

a(n) = 2^n*A001075(n).

a(n) = (1/2)*((4-2*sqrt(3))^n + (4+2*sqrt(3))^n), with n >= 0. - Paolo P. Lava, Nov 20 2008

G.f.: (1-4*x)/(1-8*x+4*x^2). - Philippe Deléham, Sep 07 2009

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-4)/(x*(3*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

MATHEMATICA

LinearRecurrence[{8, -4}, {1, 4}, 20] (* G. C. Greubel, Feb 03 2019 *)

PROG

(Sage) [lucas_number2(n, 8, 4)/2 for n in range(0, 21)] # Zerinvary Lajos, Jul 08 2008

(PARI) my(x='x+O('x^20)); Vec((1-4*x)/(1-8*x+4*x^2)) \\ G. C. Greubel, Feb 03 2019

(MAGMA) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x)/(1-8*x+4*x^2) )); // G. C. Greubel, Feb 03 2019

(GAP) a:=[1, 4];; for n in [3..20] do a[n]:=8*a[n-1]-4*a[n-2]; od; a; # G. C. Greubel, Feb 03 2019

CROSSREFS

Cf. A001075.

Sum_{k>=0} A086645(n,k)*m^k for m = 0, 1, 2, 4 gives A000007, A081294, A001541, A083884.

Sequence in context: A019482 A198630 A246021 * A106258 A085363 A275650

Adjacent sequences:  A090962 A090963 A090964 * A090966 A090967 A090968

KEYWORD

easy,nonn

AUTHOR

Philippe Deléham, Feb 29 2004

EXTENSIONS

Corrected by T. D. Noe, Nov 07 2006

STATUS

approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)