A162271 a(n) = ((5+sqrt(2))*(4+sqrt(2))^n + (5-sqrt(2))*(4-sqrt(2))^n)/2.

5, 22, 106, 540, 2836, 15128, 81320, 438768, 2371664, 12830560, 69441184, 375901632, 2035036480, 11017668992, 59650841216, 322959363840, 1748563133696, 9467073975808, 51256707934720, 277514627816448

Offset

0,1

Comments

Fourth binomial transform of A162396.

Links

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

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 22.

G.f.: (5-18*x)/(1-8*x+14*x^2).

Mathematica

LinearRecurrence[{8, -14}, {5, 22}, 50] (* G. C. Greubel, Oct 02 2018 *)
Table[((5+Sqrt[2])(4+Sqrt[2])^n+(5-Sqrt[2])(4-Sqrt[2])^n)/2, {n, 0, 20}]// Simplify (* Harvey P. Dale, May 26 2019 *)
CoefficientList[Series[(5 - 18*x)/(1 - 8*x + 14*x^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Feb 12 2026 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+r)*(4+r)^n+(5-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009
(PARI) x='x+O('x^50); Vec((5-18*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Oct 02 2018

Crossrefs

Cf. A162396.

Sequence in context: A373930 A082297 A267241 * A164593 A153789 A213167

Adjacent sequences: A162268 A162269 A162270 * A162272 A162273 A162274

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 02 2009

Status

approved