A164033 a(n) = ((4+3*sqrt(2))*(3+sqrt(2))^n + (4-3*sqrt(2))*(3-sqrt(2))^n)/4.

2, 9, 40, 177, 782, 3453, 15244, 67293, 297050, 1311249, 5788144, 25550121, 112783718, 497851461, 2197622740, 9700776213, 42821298098, 189022355097, 834385043896, 3683153777697, 16258227358910, 71767287709581

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Keywords

• nonn

Comments

Binomial transform of A020727. Third binomial transform of A164090. Inverse binomial transform of A164034.

Links

• Matthew House, Table of n, a(n) for n = 0..1542
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A020727, A164090 (2, 3, 4, 6, 8, 12), A164034.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(3+r)^n+(4-3*r)*(3-r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009

Mathematica

CoefficientList[Series[(2-3*x)/(1-6*x+7*x^2), {x, 0, 1000}],
  x] (* or *) LinearRecurrence[{6,-7},{2,9}, 50] (* G. C. Greubel, Sep 08 2017 *)

PARI

x='x+O('x^50); Vec((2-3*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 08 2017

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 2, a(1) = 9.

G.f.: (2-3*x)/(1-6*x+7*x^2).

E.g.f.: (2*cosh(sqrt(2)*x) + (3/sqrt(2))*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009