A164304 a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

3, 14, 50, 172, 588, 2008, 6856, 23408, 79920, 272864, 931616, 3180736, 10859712, 37077376, 126590080, 432205568, 1475642112, 5038157312, 17201345024, 58729065472, 200513571840, 684596156416, 2337357481984, 7980237615104

Offset

0,1

Comments

Binomial transform of A164303. Second binomial transform of A164654. Inverse binomial transform of A164305.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..185 from Vincenzo Librandi)

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

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

a(n) = ((3+4*sqrt(2))*(2+sqrt(2))^n + (3-4*sqrt(2))*(2-sqrt(2))^n)/2.

G.f.: (3+2*x)/(1-4*x+2*x^2).

E.g.f.: (3*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 13 2017

Mathematica

(3 + 2*x)/(1 - 4*x + 2*x^2) + O[x]^24 // CoefficientList[#, x]& (* Jean-François Alcover, Jun 22 2017 *)
LinearRecurrence[{4, -2}, {3, 14}, 50] (* G. C. Greubel, Sep 13 2017 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+4*r)*(2+r)^n+(3-4*r)*(2-r)^n)/2: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009
(PARI) x='x+O('x^50); Vec((3+2*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 13 2017

Crossrefs

Cf. A164303, A164654, A164305.

Sequence in context: A261043 A063025 A187917 * A098521 A084150 A203196

Adjacent sequences: A164301 A164302 A164303 * A164305 A164306 A164307

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

Status

approved